international conference on inverse problems

International Conference on Inverse Problems

April 25 - 29, 2010

Wuhan University, Wuhan

Objective

Inverse Problems has become a very active, interdisciplinary and well-established research area overthe past two decades. It has found wide applications in engineering, industry, medicine, as well as lifeand earth sciences. The aim of this conference is to bring together computational and applied math-ematicians from around the world to present and discuss recent developments in inverse problems andtheir applications. This conference will encourage international collaboration and interactive activitieson inverse problems and provide an opportunity for young researchers to learn the current state-of-the-art in the field and present their recent research results.

Organizing Committee

Hua Chen (Co-Chair), Wuhan University, ChinaHui Feng, Wuhan University, ChinaTsz Shun Chung, The Chinese University of Hong Kong, ChinaJianli Xie, Shanghai Jiao Tong University, ChinaLijuan Wang, Wuhan University, ChinaXiufen Zou, Wuhan University, ChinaJun Zou (Co-Chair), The Chinese University of Hong Kong, China

Scientific Committee

H. T. Banks, North Carolina State University, USAHeinz W. Engl, University of Vienna, AustriaWilliam Rundell, Texas A&M University, USAGunther Uhlmann, University of Washington, USAMasahiro Yamamoto, University of Tokyo, JapanJun Zou, The Chinese University of Hong Kong, Hong Kong SAR

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List of Invited Speakers

H. T. Banks, North Carolina State University, USAGang Bao, Michigan State University, USATony F. Chan, National Science Foundation, USAJin Cheng, Fudan University, ChinaHeinz W. Engl, RICAM, Austrian Academy of Sciences, AustriaVictor Isakov, Wichita State University, USAKazufumi Ito, North Carolina State University, USASergey I. Kabanikhin, Sobolev Institute of Mathematics, RussiaHyeonbae Kang, Inha University, KoreaRainer Kress, University of Goettingen, GermanyPhilipp Kuegler, RICAM, Austrian Academy of Sciences, AustriaKarl Kunisch, University of Graz, AustriaTatsien Li, Fudan University, ChinaJijun Liu, Southeast University, ChinaPeter Monk, University of Delaware, USARoland Potthast, University of Reading, UKWilliam Rundell, Texas A&M University, USAGunther Uhlmann, University of Washington, USAGengsheng Wang, Wuhan University, ChinaJenn-Nan Wang, National Taiwan University, TaiwanYuesheng Xu, Syracuse University, USA and Sun Yat-sen University, ChinaMasahiro Yamamoto, University of Tokyo, JapanBo Zhang, Chinese Academy of Sciences, ChinaXu Zhang, Chinese Academy of Sciences, China

Main Sponsors

National Natural Science Foundation of ChinaWuhan University

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CONTENTS CONTENTS

Contents

1 Programme 4

2 Invited talks 10

3 Contributed talks 21

4 List of the participants 40

5 Direction Information 47

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1 PROGRAMME

1 Programme

April 25, Sunday Registration at FengYi Hotel

April 26, Monday, Morning; Chairman: Hua Chen

08:00 - 08:45am Registration

08:45 - 09:15am Opening Ceremony

09:15 - 09:30am Photo

09:30 - 09:50am Tea Break

Chairman: Jun Zou

09:50 - 10:35am Tony F. ChanA general framework for a class of first order primal-dual algorithms forseparable convex minimization with applications to image processing

10:35 - 11:20am William RundellRational approximation methods for inverse source problems

11:20 - 12:05pm Xu ZhangUnique continuation property (UCP) of some PDEs: from the deter-ministic situation to its stochastic counterpart

12:05 - 02:00pm Lunch

April 26, Monday, Afternoon; Chairman: Tony F. Chan

02:00 - 02:45pm Karl KunischSemi-smooth Newton methods for L1 data fitting with automatic choiceof regularization parameters and noise calabriation

02:45 - 03:30pm Hyeonbae KangImaging finer details of shape and the material property of inclusionsusing higher order polarization tensors

03:30 - 03:50pm Tea Break

03:50 - 04:35pm Roland PotthastThe point source method - techniques and applications

04:35 - 05:20pm Jijun Liu

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1 PROGRAMME

Inverse Scattering Problems for Complex Obstacles

05:20 - 06:05pm Gengsheng WangAn approach to the optimal time for a time optimal control problem ofan internally controlled heat equation

06:05 - 07:30pm Dinner

April 27, Tuesday, Morning; Chairman: Heinz W. Engl

08:30 - 09:15am Gunther UhlmannCloaking and transformation optics

09:15 - 10:00am Jenn-Nan WangQuantitative uniqueness estimates for the Stokes and Lame systems


10:30 - 11:15am Kazufumi ItoNonsmooth Tikhonov regularization and semismmoth Newton method

11:15 - 12:00pm Bo ZhangUniqueness in the inverse obstacle scattering in a piecewise homoge-neous medium

12:00 - 2:00pm Lunch

April 27, Tuesday, Afternoon; Chairman: William Rundell

02:00 - 02:45pm H. T. BanksGeneralized sensitivities and optimal experimental design

02:45 - 03:30pm Peter MonkThe interior transmission problem in acoustics


Parallel Session 1A Chairman: Karl Kunisch

03:50 - 04:20pm Bangti JinParameter choice rules for sparse reconstruction

04:20 - 04:50pm Jianli Xie

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1 PROGRAMME

A posteriori error estimates of finite element methods for heat fluxreconstructions

04:50 - 05:20pm Hui FengSimultaneous identification of electric permittivity and magnetic per-meability

05:20 - 05:50pm Jingzhi LiTwo techniques to make linear sampling methods more efficient andeffective

Parallel Session 1B Chairman: Masahiro Yamamoto

03:50 - 04:20pm A. Yu. ChebotarevSubdifferential inverse problems for magnetohydrodynamics

04:20 - 04:50pm Wenyuan LiaoIdentification of acoustic coefficient of a wave equation using extraboundary measurements

04:50 - 05:20pm Lijuan WangError estimates of finite element methods for parameter identificationsin elliptic and parabolic systems

05:20 - 05:50pm Guanghui HuUniqueness in inverse scattering of elastic waves by polygonal periodicstructures

Parallel Session 1C Chairman: Rainer Kress

03:50 - 04:20pm Jonas OfftermattAn adaptive method for recovering sparse solutions to inverse problemswith applications in system biology

04:20 - 04:50pm Shitao LiuInverse problem for a structural acoustic interaction

04:50 - 05:20pm Bartosz ProtasOn optimal reconstruction of constitutive relations

05:20 - 05:50pm Na TianNumerical study to the inverse source problem with convection-diffusion equation

06:00pm Bus Departure for Banquet

06:30 - 09:00pm Banquet

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1 PROGRAMME

April 28, Wednesday, Morning; Chairman: Gunther Uhlmann

08:30 - 09:15am Heinz W. EnglInverse problems in systems biology

09:15 - 10:00am Gang BaoInverse scattering by near-field imaging


10:30 - 11:15am Sergey I. KabanikhinNumerical methods of solving inverse hyperbolic problems

11:15 - 12:00pm Victor IsakovThe increasing stability in the continuation and inverse problems forthe Helmholtz type equations

12:00 - 2:00pm Lunch

April 28, Wednesday, Afternoon

02:00 - 07:00pm Sightseeing

07:00 - 08:30pm Dinner

April 29, Thursday, Morning; Chairman: Gang Bao

08:30 - 09:15am Tatsien LiA constructive method to controllability and observability for quasilin-ear hyperbolic systems

09:15 - 10:00am Masahiro YamamotoParabolic Carleman estimates and the applications to inverse sourceproblems for equations of fluid dynamics


Parallel Session 2A Chairman: Roland Potthast

10:30 - 11:00am Eric Tse Shun ChungA new phase space method for recovering index of refraction from traveltimes

11:00 - 11:30am Serdyukov Aleksander

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1 PROGRAMME

The linearized traveltime tomography in transversal isotropic elasticmedia

11:30 - 12:00pm Igor TrooshinOn inverse scattering for nonsymmetric operators

Parallel Session 2B Chairman: Yuesheng Xu

10:30 - 11:00am Christian DaveauAsymptotic behaviour of the energy for electromagnetic systems in thepresence of small inhomogeneities

11:00 - 11:30am Shuai LuModel function approach in the modified L-curve method for the choiceof regularization parameter

11:30 - 12:00pm Dennis TredeTikhonov regularization with sparsity constraints: convergence ratesand exact recovery

Parallel Session 2C Chairman: Xu Zhang

10:30 - 11:00am Francois-Xavier Le DimetPosterior covariances of the optimal solution errors in variational dataassimilation

11:00 - 11:30am Galina ReshetovaScenario of seismic monitoring of productive reservoirs

11:30 - 12:00pm Vladimir A. TcheverdaTravel-time inversion for 3D media without of ray tracing: simultaneousdetermination of velocity and hypocenters

12:00 - 02:00pm Lunch

April 29, Thursday, Afternoon; Chairman: Peter Monk

02:00 - 02:45pm Rainer KressHuygens’ principle and iterative methods in inverse obstacle scattering

02:45 - 03:30pm Jin ChengUnique continuation on a line for Helmholtz equation

03:30 - 04:15pm Philipp KueglerApplications of sparsity enforcing regularization in systems biology


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1 PROGRAMME

Parallel Session 3A Chairman: H. T. Banks

04:35 - 05:05pm Tamas PalmaiQuantum mechanical inverse scattering problem at fixed energy: a con-structive method

05:05 - 05:35pm Vincent JugnonNumerical identification of small imperfections using dynamical meth-ods

05:35 - 06:05pm Masaaki UesakaInverse problems for some system of viscoelasticity via Carleman esti-mate

06:05 - 06:35pm Xiliang LuOptimal control for the elliptic system with general constraint

Parallel Session 3B Chairman: Hyeonbae Kang

04:35 - 05:05pm Jeff Chak-Fu WongA FE-based algorithm for the inverse natural convection problem

05:05 - 05:35pm Yunjie MaSome regularization methods for the numerical analytic continuation

05:35 - 06:05pm Wenfeng PanFast acoustic imaging for a 3D penetrable object immersed in a shallowwater waveguide

Parallel Session 3C Chairman: Jin Cheng

04:35 - 05:05pm Adel HamdiIdentification of a time-varying point source: application to rivers waterpollution

05:05 - 05:35pm Ting ZhouReconstrcting electromagnetic obstacles by the enclosure method

05:35 - 06:05pm Hui HuangEfficient reconstruction of 2D images and 3D surfaces

06:05 - 06:35pm Fenglian YangAn adaptive greedy technique for inverse boundary determination prob-lem

06:05 - 07:30pm Dinner

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2 INVITED TALKS

2 Invited talks

Generalized Sensitivities and Optimal Experimental DesignH. T. Banks

Center for Research in Scientific ComputationDepartment of Mathematics, North Carolina State University

Raleigh, NC 27695-8205Email: [email protected]

Abstract

We consider the problem of estimating a modeling parameter θ using a weighted least squares cri-terion Jd(y, θ) =

∑ni=1

1σ(ti)

2 (yi − f(ti, θ))2 for given data yi by introducing an abstract frameworkinvolving generalized measurement procedures characterized by probability measures. We take an opti-mal design perspective, the general premise (illustrated via examples) being that in any data collected,the information content with respect to estimating θ may vary considerably from one time measurementto another, and in this regard some measurements may be much more informative than others. Wepropose mathematical tools which can be used to collect data in an almost optimal way, by specifyingthe duration and distribution of time sampling in the measurements to be taken, consequently improv-ing the accuracy (i.e., reducing the uncertainty in estimates) of the parameters to be estimated. Werecall the concepts of traditional and generalized sensitivity functions and use these to develop a strategyto determine the “optimal” final time T for an experiment; this is based on the time evolution of thesensitivity functions and of the condition number of the Fisher information matrix. We illustrate therole of the sensitivity functions as tools in optimal design of experiments, in particular in finding “best”sampling distributions. Numerical examples are presented throughout to motivate and illustrate theideas. This represents joint efforts with F. Kappel, S. Dediu and S. Ernstberger.

Inverse Scattering by Near-Field ImagingGang Bao

Michigan State University & Zhejiang UniversityEmail: [email protected]

Abstract

Recent progress on inverse scattering problems for Maxwell’s equations with near-field and far-fieldboundary measurements will be presented. Our approaches could be applied to both single and multiplefrequency cases. Various analytic aspects of the approaches will be discussed. Numerical results onthe inverse problems will be presented, which are accurate, data driven, and with super-resolution fornear-field measurements. Related on-going work on inverse problems involving uncertainties will also behighlighted.

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2 INVITED TALKS

A General Framework for a Class of First Order Primal-Dual Algorithmsfor Separable Convex Minimization with Applications to Image Processing

Tony ChanThe Hong Kong University of Science and Technology

Email: [email protected]

Abstract

We generalize a recently proposed primal-dual hybrid gradient (PDHG) algorithm proposed by Zhuand Chan, draw connections to similar methods and discuss convergence of several special cases andmodifications. In particular, we point out a convergence result for a modified version of PDHG thathas a similarly good empirical convergence rate for total variation (TV) minimization problems. Itsconvergence follows from interpreting it as an inexact Uzawa method. We also prove a convergenceresult for PDHG applied to TV denoising with some restrictions on the PDHG step size parameters. Itis shown how to interpret this special case as a projected averaged gradient method applied to the dualfunctional. We discuss the range of parameters for which the inexact Uzawa method and the projectedaveraged gradient method can be shown to converge. We also present some numerical comparisons ofthese algorithms applied to TV denoising, TV deblurring and constrained l1 minimization problems.(Joint work with Ernie Esser and Xiaoqun Zhang, Math Dept, UCLA).

Unique Continuation on a Line for Helmholtz Equation1Jin Cheng, 1Xu Xiang, 2Yamamoto Masahiro

1School of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R. China2Graduate School of Mathematical Sciences, the University of Tokyo, Tokyo 163, Japan


Abstract

In this talk, local unique continuation on a line for the solution to Helmholtz equation is presented.The conditional stability estimation is proved by using complex extension method which is proposedin Cheng and Yamamoto. Since the fundamental solution of Helmholtz equation has a logarithmicsingularity which behaviors more or less like a Laplacian equation, the result obtained here can beviewed as a natural extension of the previous results. Numerical results for Laplacian and Helmholtzequations confirm the theoretical prediction of the Holder type estimation provided that the line doesnot attain the boundary of the domain.

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2 INVITED TALKS

Inverse Problems in Systems BiologyHeinz W. Engl

Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesEmail: [email protected]

Abstract

Systems biology, a relatively young biological discipline, claims to consider cells and organism asentities in a holistic way emphasizing at the same time the interplay of components from the molecularto the systemic level. One of its major goals is to reach an understanding of the properties of cellsor organisms emerging as consequence of the interaction of large numbers of molecules, which organizethemselves into highly intricate reaction networks that span various levels of complexity. There is wide-spread scepticism if or to which extent these goals are reachable. But any progress in this field dependson ”inverse problems technology”.

Inverse problems arise in at least two levels:- parameter estimation from measurements.- qualitative inverse problems that aim at the identification or reverse engineering of bifurcation

patterns and of other types of desired qualitative behaviour.In both cases, sparsity plays a major role: metabolic or genetic networks are typically sparse, and

one wants to design desired qualitative behaviour with as few changes to the system as possible.We describe regularization techniques for both types of problems and exemplify their effect by quali-

tative inverse problems in connection with the cell division cycle and the circadian rhythm, respectively.

The Increasing Stability in the Continuation andInverse Problems for the Helmholtz Type Equations

Victor IsakovDepartment of Mathematics and Statistics, Wichita State University


Abstract

We consider the Cauchy problem for the Helmholtz equation (with variable coefficients) in a do-main D with the data on Γ ⊂ ∂D and demonstrate increasing stability with growing frequency whenΩ is inside a convex hull of Γ. Proofs use energy estimates for hyperbolic equations and Carlemantype estimates. When the convexity condition is violated, the stability of all solutions is deteriorating.However for some cases we show both analytically and numerically that disregard any geometric (orpseudo-convexity) assumptions there is an increasing subspace of all solutions of the Helmholtz equationwhere the continuation problem is Lipschitz stable.

In addition, we show increasing stability of recovery of potential in the Schrodinger equation from itsDirichlet-to-Neumann map.

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2 INVITED TALKS

Nonsmooth Tikhonov Regularization and Semismmoth Newton MethodKazufumi Ito

Department of Mathematics, North Carolina State UniversityEmail: [email protected]

Abstract

Applications and analysis of the semismooth Newton method for non-smooth equations associatedwith nonsmooth Tikhonov functionals are presented. The semismmooth Newton method is a generalizedNewton method for a class of the Lipschitz but not C1 equations in Banach spaces. For the Tikhonovregularization the nonsmooth equation is reduced from the necessary optimality condition for minimizingthe cost functional involving L1, and TV (total variation) norms and/or subject to the point-wiseconstraint of the solution and its gradient. The L1 norm is used to obtain the spike and impulsivesolutions and TV norm is used to capture the edge and the discontinuity in the image. A robustalgorithm based on the primal and dual active set method and the semismooth Newton is developed andanalyzed.

Numerical Methods of Solving Inverse Hyperbolic ProblemsSergey I. Kabanikihin

Sobolev Institute of MathematicsEmail: [email protected]

Abstract

The problems of determining coefficients of hyperbolic equations and systems from some additionalinformation on their solutions are of great practical significances. We consider dynamical type of inverseproblem in which the additional information is given by the trace of the direct problem solution on a(usually time-like) surface of the domain. This kind of inverse problems were originally formulated andinvestigated by M.M. Lavrentiev and V.G. Romanov (1966). The technique developed by V.G. Romanovfor proving local theorems of unique solvability for dynamic inverse problems and also theorems ofuniqueness and conditional stability ”on the whole” was applied in the elaboration of numerical methodsfor solving a wide range of inverse problems of acoustics, siesmics, electrodynamics.

A majority of the papers and books devoted to the study of dynamic inverse problems deal with oneof the following basic methods:

- method of Volterra operator equations;- linearization and Newton-Kantorovich method;- Landweber iterations (LI) and optimization;- Gelfand-Levitan-Krein and boundary control methods;- method of finite-difference scheme inversion.The first group of methods, namely, Volterra operator equations, Newton-Kantorovich, Landweber

iteration and optimization methods produce the iterative algorithms where one should solve the corre-sponding direct (forward) problem and adjoint (or linear inverse) problem on every step of the iterativeprocess. On the contrary, the Gelfand-Levitan method, the method of boundary control, the finite-difference scheme inversion and sometimes linearization method do not use the multiple direct problemsolution and allow one to find the solution in a specific point of the medium. Therefore we will refer tothese methods as the ”direct” methods.

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2 INVITED TALKS

In the present talk we will discuss theoretical and numerical background of the direct and iterativemethods. We formulate and prove theorems of convergence, conditional stability and other properties ofthe mentioned above methods.

We will consider the following direct methods:- finite-difference scheme inversion,- linearization,- method of Gelfand-Levitan-Krein,- boundary control method,- singular value decomposition method.We intend to estimate the convergence rate of several numerical algorithms used extensively for

solving inverse problems for hyperbolic equations, be they in partial differential, integral-differential,operator, finite-difference or variational form.

Our attention is focused on inverse problems for hyperbolic equations (IPHE) for the following rea-sons. First, the great majority of IPHE can be reduced to Volterra operator equations, thus creating apossibility to extend the theoretical results and numerical methods from the Volterra equations theoryto IPHE. In particular, the Picard and Caratheodory successive approximations (PSA and CSA, respec-tively) developed for Volterra equations are applicable to IPHE. Second, a great number of numericalmethods, such as finite-difference scheme inversion, linearization and Newton-Kantorovich type methods,regularization method and optimization methods, the dynamical version of the Gelfand-Levitan methodsand many others, were elaborated for and tested on IPHE. A fair amount of experimental results hasbeen accumulated, thereby inducing researchers to systematize and generalize the compiled material.Finally, it is well known that in some important cases the direct problems for elliptic and parabolicequations reduce to those for hyperbolic ones. Therefore, the technique worked out for IPHE may alsobe useful in studying inverse problems for elliptic or parabolic equations.

We formulate inverse problems in the form of operator equation Aq = f and in variation form

J(q) =< Aq − f,Aq − f > → min .

In direct methods we construct the solution without solving the corresponding direct (forward) prob-lem. We describe linearization: A′(q0)q1 = f1, finite-difference scheme inversion, Gelfand-Levitan andboundary control methods, and SVD algorithms.

In iterative methods it is necessary to solve direct problem and corresponding conjugate or inverseproblem. We consider Landweber iteration

qn+1 = qn − α[A′(qn)]∗(Aqn − f),

gradient methodsqn+1 = qn − αnJ ′(qn), J ′(q) = 2[A′(q)]∗(Aq − f),

Newton-Kantorovich methodqn+1 = qn − [A′(qn)]−1(Aqn − f).

We analyze the convergence and the rate of convergence of direct and iterative methods using thesimplest examples of inverse problems of acoustics, electrodynamics and heat transfer.

The work was supported by RFBR grant No 09-01-00746-a.

References

[1] S. I. Kabanikhin, Inverse and Ill-Posed Problems, Siberian Science Publishers, Novosibirsk, 2009, 456 pp.[2] S. I. Kabanikhin, Definitions and examples of inverse and ill-posed problems, Journal of Inverse and Ill-PosedProblems, 2008, 16(4), pp. 317-357.

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[3] S. I. Kabanikhin, A. D. Satybaev, M. A. Shishlenin, Direct Methods of Solving Inverse Hyperbolic Problems,2004, VSP/BRILL, the Netherlands, 179 pp.

Imaging Finer Details of Shape and the Material Property ofInclusions Using Higher Order Polarization Tensors

Hyeonbae KangDepartment of Mathematics, Inha University


Abstract

It is well-known that using the polarization tensor we can recover the “equivalent ellipse (ellipsoid)” of theinclusions. The equivalent ellipse is an ellipse which represents the overall property of the inclusion. In this talkwe will discuss on possibility of using higher order polarization tensors to recover finer details of the shape of theinclusion and to separate the material property, such as conductivity and stiffness, from the volume.

This talk is based on joint works with H. Ammari, E. Kim, J.-Y. Lee, M. Lim, and H. Zribi.

Huygens’ Principle and Iterative Methods in Inverse Obstacle ScatteringRainer Kress

Institut fur Numerische und Angewandte MathematikUniversitat Gottingen, Germany


Abstract

The inverse problem we consider is to determine the shape of an obstacle from the knowledge of the farfield pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle,we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, forthe unknown boundary of the scatterer and the induced surface flux. Reflecting the ill-posedness of the inverseobstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknownflux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilitiesfor their iterative solution via linearization and regularization.

We will first discuss the mathematical foundations of these algorithms including results on injectivity anddense range for the linearized operators and then describe the main ideas of their numerical implementation.Further, we will illuminate various relations between these three solution methods and exhibit connections anddifferences to the traditional regularized Newton type iterations as applied to the boundary to far field map.Numerical results in 3D are presented.

This is joint work with Olha Ivanyshyn, Gottingen, Germany, and Pedro Serranho, Coimbra, Portugal.

References

[1] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Springer, Berlin1998.[2] O. Ivanyshyn, R. Kress, P. Serranho, Huygens’ principle and iterative methods in inverse obstacle scattering,Advances in Computational Mathematics, DOI 10.1007/s10444-009-9135-6.

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Applications of Sparsity Enforcing Regularization in Systems BiologyPhilipp Kuegler

Johann Radon Institute for Computational and Applied MathematicsEmail: [email protected]

Abstract

One postulate of systems biology is that diseases arise from disease-perturbed biological networks. In thepursuit of understanding diseases by a combination of experimental and computational techniques also mathe-matical models of biochemical reaction networks find their applications. Focusing on ODE type models we giveexamples of how such models are built or analyzed by means of sparsity promoting regularization methods. Forsolving underdetermined systems of nonlinear equations we present an algorithm that aims at sparse correctionsof the initial guess. It is applied to an inverse bifurcation analysis of a model for intrinsic signaling pathways ofapoptosis.

Semi-smooth Newton Methods for L1 Data Fitting with AutomaticChoice of Regularization Parameters and Noise Calabriation

Karl KunischInstitute of Mathematics, Scientific Computing, University of Graz


Abstract

Inverse problems with L1 fidelity terms are investigated. Using convex analysis techniques the non-differentiableproblem is reformulated as smooth optimization problem with pointwise constraints. This can be efficiently solvedusing semismooth Newton methods. In order to achieve superlinear convergence, the dual problem requires ad-ditional regularization, which is based on the structure of the problem. Regularization in the primal problemis chosen according to a balancing principle derived from the model function approach, which does not rely onknowledge of the noise level in the data.

This is joint work with C. Clason and B. Jin.

A Constructive Method to Controllability andObservability for Quasilinear Hyperbolic Systems

Tatsien LiSchool of Mathematical Sciences, Fudan University


Abstract

In this talk a simple constructive method is presented to get the exact boundary controllability, the exactboundary controllability of nodal profile and the exact boundary observability for 1-D quasilinear hyperbolicsystems.

Inverse Scattering Problems for Complex ObstaclesJijun Liu

Department of Mathematics, Southeast UniversityEmail: [email protected]

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2 INVITED TALKS

Abstract

For given incident wave, the scattered wave outside an obstacle is governed by the Helmholtz equation infrequency domain. The inverse scattering problems aim to detect the obstacle property such as boundary shapeand type from some information about the scattered wave. Such kind of inverse problems have been studiedthoroughly using optimization techniques from the numerical points of view. In this talk, we will introducesome recent works for inverse scattering problems by complex obstacles. By complex obstacle, we mean that theobstacle may be of different acoustic wave property on the different part of the boundary, or the obstacle is acrack. We will present the reconstruction behavior of singular source method and show the effect of boundaryimpedance and boundary curvature.

The Interior Transmission Problem in AcousticsPeter Monk

Department of Mathematical Sciences, University of DelawareEmail: [email protected]

Abstract

As a result of studies of the far field pattern of the scattered wave for time harmonic acoustic waves, a newclass of interior problem arises termed the ”Interior Transmission Problem” (ITP). The ITP is not a standardelliptic problem, and a study of the solvability of this problem gives rise to a non- standard eigenvalue problem forthe ITP. The proof of existence and properties of these eigenvalues is not straightforward. I shall survey the ITPand its properties in several applications, and describe numerical schemes for computing transmission eigenvalues.Remarkably, transmission eigenvalues can be observed from far field data, and the resulting eigenvalues can beused to estimate properties of the scatterer. The interior transmission problem has similarities with problemsinvolving negative index of refraction, so it may be that the study of interior transmission eigenvalues is alsorelevant there.

The Point Source Method - Techniques and ApplicationsRoland Potthast

German Meteorological Service - Deutscher WetterdienstInst. Num. Appl. Mathematics, University of Gottingen

Department of Mathematics, University of ReadingEmail: [email protected]

Abstract

We will provide a survey about the point source method for the reconstruction of acoustic, electromagneticor fluid dynamical fields from remote measurements. The method is based on an application of Green’s theoremor related formulas for different applications to represent fields by surface integrals. An approximation of thefundamental solutions, which are typically involved in the representation, by a superposition of incident fieldsleads to a reconstruction formula to calculate the fields from remote measurements. The method can be usedfor various application areas. It provides a very effective tool which is employed in probing methods for objectreconstruction. We will discuss further applications for source splitting and source identification.

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2 INVITED TALKS

Rational Approximation Methods for Inverse Source ProblemsWilliam Rundell

Department of Mathematics, Texas A&M UniversityEmail: [email protected]

Abstract

The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one canonly make measurements on an exterior surface. Such is the ubiquity of these problems, the underlying modelcan lead to a partial differential equation of any of the major types, but here we focus on the case of steady-stateelectrostatic or thermal imaging and consider boundary value problems for Laplace’s equation. Our inclusionsare interior forces with compact support and our data consists of a single measurement of (say) voltage/currentor temperature/heat flux on the external boundary. We propose an algorithm that under certain assumptionsallows for the determination of the support set of these forces by solving a simpler “equivalent point source”problem.

Cloaking and Transformation OpticsGunther Uhlmann

Department of Mathematics, University of WashingtonEmail: [email protected]

Abstract

We describe recent theoretical and experimental progress on making objects invisible to detection by electro-magnetic waves, acoustic waves and matter waves. For the case of electromagnetic waves, Maxwell’s equationshave transformation laws that allow for design of electromagnetic materials that steer light around a hiddenregion, returning it to its original path on the far side. Not only would observers be unaware of the contents ofthe hidden region, they would not even be aware that something was being hidden. The object, which wouldhave no shadow, is said to be cloaked. We recount some of the history of the subject and discuss some of themathematical issues involved.

An Approach to the Optimal Time for a Time Optimal ControlProblem of an Internally Controlled Heat Equation

Gengsheng Wang, Guojie ZhengDepartment of Mathematics, School of Mathematics and Statistics, Wuhan University


Abstract

In this paper, we study the optimal time for a time optimal control problem (P), where the state systemis an internally controlled heat equation. By projecting the original problem via the finite element method, weobtain another time optimal control problem (Phl) governed by a linear system of ordinary differential equations.Here, h and l are the mesh sizes of the finite element spaces of the state space and the control space respectively.The purpose of this study is to find the relationship between the optimal times for the original and projectedproblems. We prove that the optimal time for the problem (Phl) converges to the optimal time for the problem(P) when h and l are approaching zero. More significantly, we obtain error estimates between the optimal timesin terms of h and l from the following two cases: (i) the state system of (P) is the globally controlled heatequation; (ii) the state system of (P) is an internally controlled one-dimensional heat equation. Also, we derive

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that the projected internally controlled one-dimensional heat equation (via the finite element method) is exactlycontrollable.

Quantitative Uniqueness Estimates for the Stokes and Lame SystemsJenn-Nan Wang

Department of Mathematics, National Taiwan UniversityEmail: [email protected]

Abstract

In this talk, I would like to discuss quantitative uniqueness estimates for the Stokes and Lame systems withsingular and Lipschitz coefficients, respectively. Due to insufficient smoothness of the coefficients, both systemscan not be decoupled. We will prove our results by the Carleman method. Our quantitative uniqueness resultsimply the strong unique continuation property. For the Lame system with Lipschitz coefficients, the result isoptimal.

Fast Algorithms for Inverting the RegularizedKernel Matrix in Machine Learning

1Guohui Song, 2Yuesheng XuIllinois Institute of Tech, Syracuse University & Sun Yat-sen University

1Email: [email protected], 2Email: [email protected]

Abstract

Many kernel based machine learning methods require to invert the regularized kernel matrix. Normally,such a kernel matrix is a full matrix of large size. Especially, for high dimensional data, it is difficult to invertthe regularized kernel matrix. This is a bottleneck problem of the implementation of the kernel based machinelearning methods. To efficiently invert the regularized matrix, we approximate the kernel matrix by a multilevelcirculant matrix and then apply the super-fast Fourier transform to the regularized multilevel circulant matrixto find its inverse. Theoretical results and numerical examples will be presented in the talk.

Parabolic Carleman Estimates and the Applications toInverse Source Problems for Equations of Fluid Dynamics

Masahiro YamamotoDepartment of Mathematical Sciences, University of Tokyo


Abstract

First we derive a Carleman estimate for a parabolic equation by a direct method. Next we will apply theCarleman estimate to inverse problems of determining source terms in the Navier-Stokes equations and relatedequations and we establish conditional stability estimates.

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Uniqueness in the Inverse Obstacle Scatteringin a Piecewise Homogeneous Medium

X. Liu, B. ZhangLSEC and Institute of Applied Mathematics

Academy of Mathematics and Systems ScienceChinese Academy of Sciences, Beijing 100190


Abstract

In this talk, we are concerned with the inverse scattering problem by an impenetrable obstacle embeddedin a piecewise homogeneous medium. We will establish a uniqueness result in determining both the penetrableinterface between the layered media and the impenetrable obstacle with its physical property from a knowledgeof the far field pattern for incident plane waves. This we do by establishing both a new mixed reciprocity relationand a priori estimates of the solution on some part of the interface for the direct problem, employing the integralequation method.

Unique Continuation Property (UCP) of Some PDEs:from the Deterministic Situation to its Stochastic Counterpart

Xu ZhangAcademy of Mathematics and Systems Science, Chinese Academy of Sciences


Abstract

In this talk, I will recall some classical results/tools on the UCP of deterministic PDEs, and explain whythe study of the global UCP, even when the local UCP is not clear or even negative, is possible and necessary.Further, I will analyze the difficulty and new phenomenon appeared in the study of UCP for stochastic PDEs,and especially, why the problem in the stochastic setting has to be considered globally in some sense. Finally, Iwill present some new UCP results for stochastic PDEs and comment a list of open problems.

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3 Contributed talks

The Linearized Traveltime Tomographyin Transversal Isotropic Elastic Media

Serdyukov AleksanderNovosibirsk State University, Russia


Abstract

The standard method of seismic data processing is to introduce the velocity model as the superposition of twocomponents: the smooth macro velocity model that doesn’t practically change the wave propagation direction andthe high contrast component that has no influence on traveltimes but changes propagation directions (reflectingand scattering objects). The presence of elastic anisotropy produces extra demands to macrovelocity model. Thatis why our study is concerned with the macrovelocity model recovering in the anisotropic media. We considertomography problem in transversal isotropic media (TI media).

Before solving tomography problem it is necessary to find out the possibility of the different anisotropicparameters recovering. That is why our study is devoted to the analyses: we consider qP traveltimes in TI mediaand study what parameters can be recovered and which way of the TI media parameterization is the optimalone for the tomography.

Let’s search the observed elastic TI media parameters m as a sum of two components m = m0 + dm, wherem0 are the parameters of the undisturbed ”containing” media expected to be known, and dm are their smalldisturbances to be recovered using observed qP-wave traveltimes. These assumptions lead to linearized statementof the tomography problem that can be formally represented by equation:

B < dm >= dτ, (1.1)

where dτ are time residuals between computed traveltimes in ”containing” media (with known parameters m0)and real observed traveltimes for every pair source-receiver, B is linear integral tomography operator.

In practice the equation (1.1) is solved numerically and instead of linear integral operator B its matrixapproximation B is used. The heart of the problem is the solution of a linear system of equations. Matrix Binherits the properties of the initial operator B. B is compact operator so matrix B is ill-conditioned (has verylarge condition number). SVD was performed on the matrix B for the different parameterization. To understandwhat parameters can be resolved it is necessary to find out the structure of stable decisions spaces. These spacesare linear shells of the highest right singular vectors. If the order of the data error is know the error in solutioncan be controlled by the number of involved right singular vectors .

We have find out that optimal parameterization for linearized inverse kinematic problem is Schoenberg’sparameterization. Only two of three parameters can be practically defined: p wave velocity along the axis ofsymmetry direction and ellipticity parameter.

Subdifferential Inverse Problems for MagnetohydrodynamicsA. Yu. Chebotarev

Department of Mathematical Modeling, Institute for Applied MathematicsFEB Russian Academy of Science, 7 Radio St., 690041 Vladivostok, Russia


Abstract

The magnetohydrodynamic (MHD) equations in the bounded domain Ω ⊂ Rd with the boundary Γ areconsidered.

∂u/∂t− ν∆u+ (u∇)u = −∇p+ S · rotB ×B, x ∈ Ω, t > 0, (1)

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∂B/∂t+ rotE = 0, rotB = 1/νm(E + u×B + Ec), div u = 0, divB = 0. (2)

Here u, B, E and Ec =∑m

i=1 αi(t)Ei are vector fields of velocity, magnetic induction, electric intensity andexternal electromotive intensity respectively; p is the flow pressure, ν = 1/Re. νm = 1/Rm, S = M2/ReRm,where Re,Rem and M are the Reynolds number, Reynolds magnetic number and Hartmann number.

To the equations (1)-(2) we add the initial and the boundary value conditions

u|t=0 = u0(x), B|t=0 = B0(x), x ∈ Ω, u|Γ = 0, B · n|Γ = 0, n× E|Γ = 0, (3)

where n is the unit outward normal to the boundary.The following inverse problem is considered. Find the functions αi = αi(t), i = 1, ...,m and the solution

u,B of (1)-(3) satisfying the additional conditions

αi(t) ≥ 0,

∫Ω

rotB · Ei dx ≥ qi(t), αi(t)(

∫Ω

rotB · Ei dx− qi) = 0. (4)

The functions qi(t), Ei(x) are given.To study the Problem (1)–(4) the theory of solvability of an evolution inequality in a Hilbert space for the

operators with the quadratic nonlinearity is created. Obtained results is used for the study of MHD flows. Forthe 3 - dimensional flows the global in time existence of the weak solutions to the variational inequalities isproved. For the two-dimensional flows existence and uniqueness of the strong solutions are proved. Then theinverse problem is considered. On the base of estimates of the solution for subdifferential problem for the MHDsystem the solvability is proven on the condition that the norm of Ec is minimal.

A New Phase Space Method for RecoveringIndex of Refraction from Travel Times

Tsz Shun Eric ChungDepartment of Mathematics, The Chinese University of Hong Kong


Abstract

We develop a new phase space method for reconstructing the index of refraction of a medium from travel timemeasurements. The method is based on the so-called Stefanov-Uhlmann identity which links two Riemannianmetrics with their travel time information. We design a numerical algorithm to solve the resulting inverseproblem. The new algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations.In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally,while the Eulerian formulation for the index of refraction allows us to compute the solution in physical space.Numerical examples including isotropic metrics and the Marmousi synthetic model are shown to validate thenew method.

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Asymptotic Behaviour of the Energy for ElectromagneticSystems in the Presence of Small Inhomogeneities

1Christian Daveau, 2Abdessatar Khelifi1Department de Mathematiques, CNRS AGM UMR 8088,

Universite de Cergy-Pontoise, 95302 Cergy-Pontoise Cedex, France2Department de Mathematiques,

Universite des Sciences de Carthage, Bizerte TunisieEmail: [email protected]

Abstract

For such solution, of the time-harmonic case, we give a rigorous derivation of the asymptotic expansions in theinteresting situation when a finite number of inhomogeneities of small diameter are embedded in the entire space.Then, we analyze the behavior of the electromagnetic energy caused by the presence of these inhomogeneities.In case of general time dependent, we show that the local electromagnetic energy, trapped in the total collectionof these well separated inhomogeneities, decays toward zero as the shape parameter decreases to zero or as timeincreases.

References

[1] H. Ammari, E. Iakovleva, D. Lesselier, G. Perrusson, MUSIC-type electromagnetic imaging of a collection ofsmall three-dimensional inclusions, SIAM Sci. Comput., 29 (2007), pp. 674-709.[2] H. Ammari, M. S. Vogelius, D. Volkov, Asymptoticc formulas for perturbations in the electromagnetic fieldsdue to the presence of inhomogeinities of small diameter II. The full Maxwell equations, J. Math. Pures Appl.,80 (2001), pp. 769-814.[3] B. Ducomet, Decay of Electromagnetic Energy in a Perturbed Half-Space, Computers Math. Applic., 29(1995), No. 3, pp. 13-21.[4] M. Schiffer, G. Szego, Virtual mass and polarization, Trans. Amer. Math. Soc., 67 (1949), pp. 130-205.[5] M. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to thepresence of inhomogeneities, Math. Model. Numer. Anal., 34 (2000), pp. 723-748.

Posterior Covariances of the Optimal SolutionErrors in Variational Data Assimilation

F. -X. Le Dimet1, I. Gejadze2,V. Shutyaev3

1MOISE project (CNRS, INRIA, UJF, INPG); LJK, Universite de GrenobleBP 51, 38051 Grenoble Cedex 9, France,

2Department of Civil Engineering, University of Strathclyde,John Anderson Building, 107 Rottenrow, Glasgow, G4 ONG, UK

3Institute of Numerical Mathematics, Russian Academy of Sciences,119333 Gubkina 8, Moscow, Russia

1Email: [email protected]

Abstract

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimalcontrol problem to find unknown model parameters such as initial and/or boundary conditions, right-hand sides(forcing), and distributed coefficients. A necessary optimality condition reduces the problem to the optimalitysystem (see, e.g., [1]) which includes input errors (background and observation errors); hence the error in theoptimal solution. The error in the optimal solution can be derived through the errors in the input data usingthe Hessian of the cost functional of an auxiliary data assimilation problem. For a deterministic case it wasdone in [2]. In [3], a similar result was obtained for the continuous operator formulation, where a nonlinearevolution problem with an unknown initial condition was considered, when errors in the input data are random

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and subjected to the normal distribution. In [4], we presented an extension of the results reported in [3] forthe case of other model parameters (boundary conditions, coefficients, etc.) and show that in a nonlinear casethe a posteriori covariance operator of the optimal solution error can be approximated by the inverse Hessianof the auxiliary data assimilation problem based on the tangent linear model constraints. Here we investigatecases of highly non-linear dynamics, when the inverse Hessian does not properly approximate the analysis errorcovariance matrix, the latest being computed by the fully non-linear ensemble method with a significant ensemblesize. A modification of this method that allows us to obtain sensible approximation of the covariance with amuch smaller ensemble size is presented. Numerical examples are presented for a nonlinear convection-diffusionproblem and Burgers equation.

References

[1]F. -X. Le Dimet, O. Talagrand, Variational algorithms for analysis and assimilation of meteorological obser-vations: theoretical aspects, Tellus, 1986, v. 38A, pp. 97-110.[2] F. -X. Le Dimet, V. Shutyaev, On deterministic error analysis in variational data assimilation, NonlinearProcesses in Geophysics, 12 (2005), pp. 481-490.[3] I. Gejadze, F. -X. Le Dimet, V. Shutyaev , On analysis error covariances in variational data assimilation,SIAM J. Sci. Computing, 30 (2008), no. 4, pp. 1847-1874.[4] I. Gejadze, F. -X. Le Dimet, V. P. Shutyaev, On optimal solution error covariances in variational dataassimilation problems, J. Comp. Phys., 2009, doi:10.1016/j.jcp.2009.11.028.

Simultaneous Identification of ElectricPermittivity and Magnetic Permeability

Hui FengSchool of Mathematics and Statistics, Wuhan University


Abstract

In this paper we will investigate the simultaneous reconstruction of the electric permittivity and magneticpermeability. The two physical parameters are allowed to be highly discontinuous in the concerned physicaldomain. The ill-posed inverse problem is formulated into an output least-squares nonlinear minimization withBV-regularization. The regularizing effect and mathematical properties of the regularized system are justifiedand analysed. A fully discrete Nedelec’s edge element method is applied to approximate the regularized nonlinearoptimization system, and its convergence is demonstrated. This is a joint work with Daijun Jiang and Jun Zou.

Identification of a Time-varying Point Source:Application to Rivers Water Pollution

Adel HamdiINSA de Rouen, France


Abstract

We are interested in the inverse source problem that consists of the localization of a sought point source Sand the recovery of the history of its time-dependent intensity function λ which are both unknown and involvedin the following system of two coupled 1D advection-diffusion-reaction equations:

L[u](x, t) = λ(t)δ(x− S) for 0 < x < `, 0 < t < T,L[v](x, t) = Ru(x, t) for 0 < x < `, 0 < t < T,u(x, 0) = v(x, 0) = 0 for 0 < x < `,u(0, t) = v(0, t) = 0and u(`, t) = v(`, t) = 0 for 0 < t < T,

(1)

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where L[w](x, t) =: ∂tw(x, t)−D∂xxw(x, t) + V ∂xw(x, t) +Rw(x, t).A motivation of our study is an environmental application where the aim is to identify pollution sources in

rivers. Therefore, the system (1) corresponds to the coupled u = [BOD] (Biological Oxygen Demand) concen-tration and v = [OD] (Oxygen Deficit) concentration model, see [3,4,5]. The inverse source problem with whichwe are concerned here consists of the determination of the elements (S, λ) defining the sought pollution sourceusing some records of the OD concentration. In [2], we treated the identification of (S, λ) where only the firstequation in (1) is considered. That corresponds to the identification of the sought pollution source using somerecords of the BOD concentration.

In [1], assuming that the time-dependent intensity function λ vanishes before reaching the final control timeT , we proved the identifiability of the sought point source using the records of the state v in (1) at two interiorobservation points. Note that the two observation points should frame the source region and at least one of themshould be strategic. We established an identification method that uses the records of v to identify the sourceposition S as the root of a continuous and strictly monotonic function. Whereas the source intensity function λis recovered using a recursive formula without any need of an iterative process. Some numerical experiments on avariant of the surface water pollution BOD−OD coupled model are presented and compared to those obtainedin [2].

References

[1] A. Hamdi, Identification of a time-varying point source in a system of two coupled linear diffusion-advection-reaction equations: application to surface water pollution, Inverse Problems, Vol. 25, No 11, 115009, 2009.[2] A. Hamdi, The recovery of a time-dependent point source in a linear transport equation: application to surfacewater pollution, Inverse Problems, Vol. 25, No 7, 075006, 2009.[3] C. Linfield et al, The enhanced stream water quality models QUAL2E and QUAL2E-UNCAS: Documentationand user manuel, EPA: 600/3-87/007, May 1987.[4] Okubo, Diffusion and Ecological Problems: Mathematical Models, Springer-Verlag, New York, 1980.[5] J. S. F. Roldao, J. H. P. Soares, L. C. Wrobel, T. R. Buge, N. L. C. Dias, Pollutant Transport Studies in theParaiba Do Sul River Brazil, Water Pollution, pp. 167-180, 1991.

Uniqueness in Inverse Scattering of ElasticWaves by Polygonal Periodic Structures

Guanghui HuWeierstrass Institute for Applied Analysis and Stochastics,

Mohrenstr. 39, 10117 Berlin, GermanyEmail: [email protected]

Abstract

This talk is concerned with the inverse scattering of a time-harmonic elastic wave by an unbounded periodicstructure of R2. Such structures are also called diffraction gratings and have many important applications indiffractive optics, radar imaging and non-destructive testing.

We assume that a periodic surface divides the three-dimensional space into two non-locally perturbed half-spaces filled with homogeneous and isotropic elastic media. Moreover, we suppose that this surface is periodic inx1-direction and invariant in x3-direction, and that all elastic waves are propagating perpendicular to the x3-axisin the upper half-space. This special geometry implies that the problem can be treated as a problem of planeelasticity. Furthermore, the diffraction grating is supposed to have an impenetrable surface on which normalstress and tangential displacement (or normal displacement and tangential stress) vanish. This gives rise to theso-called the third (or fourth) kind of boundary conditions for the Navier equation. The direct problem is topredict the displacement distribution given the incident elastic waves and grating profile, whereas the inverseproblem is to determine the grating profile from a knowledge of incident waves and the near-field measurementson a straight line above the grating.

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In general, global uniqueness for the inverse problem is impossible by taking the measurements from a fixednumber of incident pressure or shear waves, which can be seen from the scattering by flat gratings. We investigatethe global uniqueness by several incident waves within an admissible class of grating profiles A, which consists ofthe non-flat graphs given by piecewise linear functions. Based on the reflection principle for the Navier equationdeveloped by Elschner J. and Yamamoto M. [2009] and a novel idea established by Bao G., Zhang H. and ZouJ., we deduce that the total field remains rotation-invariant and only consists of the compressional (or shear)part for the incident pressure (or shear) wave. The rotational invariance together with the form of incident waveleads to explicit representations of the total field as well as special geometrical structures of the grating profile.This enable us to classify all the grating profiles of A that can not be identified by one incident elastic waveand thus to prove global uniqueness with a mimimal number of incident elastic waves in the resonance case.Compared to the case where Rayleigh frequencies are exluded, the resonace case gives rise to more classes ofunidentifiable gratig profiles, which provide non-uniqueness examples for appropriately chosen wave number andincident angles.

This is a joint work with Johannes Elschner.

Efficient Reconstruction of 2D Images and 3D SurfacesHui Huang

Department of Mathematics and Earth and Ocean Sciences,University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada


Abstract

The goal of this talk is to introduce several effective techniques for solving inverse problems arising from 2Dimage and 3D surface reconstruction. Both computational and theoretical issues will be discussed.

The first part of the talk is concerned with the recovery of 2D images, e.g., denoising and deblurring. We firstconsider implicit methods that involve solving linear systems at each iteration. An adaptive Huber regularizationfunctional is used to select the most reasonable model and a global convergence result for lagged diffusivity isproved. Two mechanisms—multilevel continuation and multigrid preconditioning—are proposed to improve effi-ciency for large-scale problems. Next, explicit methods involving the construction of an artificial time-dependentdifferential equation model followed by forward Euler discretization are analyzed. A rapid, adaptive scheme isthen proposed, and additional hybrid algorithms are designed to improve the quality of such processes. We alsodevise methods for more challenging cases, such as recapturing texture from a noisy input and deblurring animage in the presence of significant noise.

It is well-known that extending image processing methods to 3D triangular surface meshes is far from trivialor automatic. In the second part of the talk we discuss techniques for faithfully reconstructing such surfacemodels with different features. Some models contain a lot of small yet visually meaningful details, and typicallyrequire very fine meshes to represent them well; others consist of large flat regions, long sharp edges (creases)and distinct corners, and the meshes required for their representation can often be much coarser. All of thesemodels may be sampled very irregularly. For models of the first class, we methodically develop a fast multiscaleanisotropic Laplacian (MSAL) smoothing algorithm. To reconstruct a piecewise smooth CAD-like model in thesecond class, we design an efficient hybrid algorithm based on specific vertex classification, which combines K-means clustering and geometric a priori information. Hence, we have a set of algorithms that efficiently handlesmoothing and regularization of meshes large and small in a variety of situations.

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Parameter Choice Rules for Sparse ReconstructionBangti Jin

Center for Industrial Mathematics, University of BremenD-28334, Bremen, Germany


Abstract

Recently, minimization problems involving so-called sparsity constraints, e.g. l1 regularization, have receivedmuch attention. However, the crucial problem of choosing an appropriate regularization parameter remainslargely unexplored. In this talk we shall discuss some rules for choosing regularization parameters in l1 regular-ization. Firstly, we revisit classical Morozov’s discrepancy principle. Theoretical properties of the discrepancyequation, e.g. existence, uniqueness, error estimates and bounds, are investigated. Its efficient numerical realiza-tion is also briefly discussed. Secondly, we adapt some heuristic rules, e.g. balancing principle and generalizedcross validation, and numerically evaluate their utility.

Numerical Identification of Small Imperfections Using Dynamical Methods1Mark Asch, 2Vincent Jugnon

1LAMFA, CNRS UMR 6140, Universite de Picardie Jules Verne, 33 rue St Leu, 80039 Amiens, France2Centre de Mathematiques Appliquees, CNRS UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France

1Email: [email protected], 2Email: [email protected]

Abstract

The imaging of small imperfections has been studied, in various contexts, using static boundary measurements(see the monograph [1] and references therein) and transient wave boundary measurements (see [2], [3] and [4]).In the limited-view case, where measurements are available on only a part of the object’s boundary, the theoryis posed in [5] and numerical computations were performed in [6]. These computations were able to validate thegeneral algorithmic approach that is based on the exact controllability of the wave equation [7] and geometriccontrol theory [8]. This study proposes an extension of the numerical work to the case of the photoacousticequations, where energy absorption of an optical pulse causes thermo-elastic expansion of the tissue, which inturn leads to propagation of a pressure wave. This pressure wave can then be measured by transducers distributedon the boundary of an organ.

Let Ω be a bounded, smooth subdomain of Rd, d = 2, 3, with for simplicity, a smooth boundary ∂Ω. Wesuppose that Ω contains a finite number m of imperfections, each of the form zj + αBj , where Bj ⊂ Rd isa bounded, smooth domain containing the origin. This gives a collection of imperfections of the form Bα =∪m

j=1(zj + αBj). The points zj ∈ Ω, j = 1, ...,m that define the locations of the imperfections are assumedto satisfy some separation conditions. Let γ0 denote the conductivity of the background medium which, forsimplicity, we shall assume is constant. Let γj be the constant conductivity of the j-th imperfection, zj + αBj .We define the piecewise constant conductivity

γα(x) =

γ0, ifx ∈ Ω\Bα

γj , ifx ∈ zj + αBj .

Consider the initial boundary value problem for the wave equation, in the presence of the imperfections,∂2uα

∂t2−∇ · (γα∇uα) = 0 in Ω× (0, T ),

uα = f on ∂Ω× (0, T ),

uα|t=0 = u0 ,∂uα

∂t

∣∣∣∣t=0

= u1 in Ω.

(1)

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If we suppose that the measurements are only done on a part of the boundary, then the detection of the absorbersfrom these partial measurements hold only under an extra assumption on T and the support of the control.Geometric control theory can be used to construct an appropriate probe function in this limited-view data case.

This problem is solved numerically by a finite-element discretization in space, using P1 conforming elements,and a Newmark method in time. The control problem is solved using a bi-grid implementation of the HUMalgorithm [9]. The actual identification is performed by a Fourier method in the pure conductivity case and byan arrival-time algorithm in the photoacoustic case. We will present some new numerical results for both cases.

References

[1] H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, Math and Applications,Volume 62, Springer-Verlag, Berlin, 2008.[2] H. Ammari, E. Bossy, V. Jugnon, and H. Kang, Mathematical modelling in photo-acoustic imaging, SIAMReview (2010), to appear.[3] H. Ammari, Y. Capdeboscq, H. Kang, and A. Kozhemyak, Mathematical models and reconstruction methodsin magneto-acoustic imaging, Euro. J. Appl. Math., 20 (2009), pp. 303–317.[4] H. Ammari, P. Garapon, L. Guadarrama Bustos, H. Kang, Transient anomaly imaging by the acoustic radiationforce, J. Diff. equat. (2010), to appear.[5] H. Ammari, An inverse initial boundary value problem for the wave equation in the presence of imperfectionsof small volume, SIAM J. Control Optim., 41 (2002), pp. 1194-1211.[6] M. Asch, M. Darbas, J.-B. Duval, Numerical solution of an inverse initial boundary value problem for thewave equation in the presence of conductivity imperfections of small volume, ESAIM Control Optim. Calc. Var.,(2010), to appear.[7] J.-L. Lions, Controabilite exacte, Perturbations et Stabilisation de Systemes Distribues, Tome 1, Controabiliteexacte, Masson, Paris, 1988.[8] C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control and stabilization ofwaves from the boundary, SIAM J. Control Optim., 30 (1992), pp. 1024-1065.[9] M. Asch and G. Lebeau, Geometrical aspects of exact boundary controllability for the wave equation - Anumerical study, ESAIM: control, Optimization and Calculus of Variations, 3 (1998), pp. 163-212.

Two Techniques to Make Linear Sampling MethodsMore Efficient and Effective

Jingzhi LiSwiss Federal Institute of Technology Zurich

Ramistrasse 101, CH-8092, Zurich, SwitzerlandEmail: [email protected]

Abstract

In this presentation, we will discuss in detail about two techniques to make linear sampling methods efficientand effective. First, we introduce the inverse obstacle scattering problems, to which the methods will apply.Existing theoretical results and numerical methods are reviewed. Among them, we will focus on linear samplingmethods (LSM), which essentially exploits the blow-up behavior of the indicator functioin around the boundariesof the obstacles on some sampling mesh. Two techniques are presented to enhance the LSM, i.e., Multilevel andstrengthened ideas. For the Multilevel LSM, it is proved that it possesses asymptotic optimal computationalcomplexity, while for the strengthened LSM, a practical guide is developed for the choice of cut-off values withresort to any known obstacle component as well as a by-product, namely the fact that interior eigenvalue problemsdo not exist when there are noise in data. Some conclusions are made in the end. This is a joint work withHongyu Liu and Jun Zou.

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Identification of Acoustic Coefficient of a Wave EquationUsing Extra Boundary Measurements

Wenyuan LiaoDepartment of Mathematics and Statistics, University of Calgary,

2500 University Drive NW, Calgary, Alberta, CanadaEmail: [email protected]

Abstract

In this paper we apply the adjoint-based optimization technique to estimate the acoustic coefficients of awave equation. Various types of acoustic coefficients, constant, time-dependent and space-dependent acousticcoefficients are investigated in this paper. The forward problem is numerically solved by some efficient and accu-rate finite difference methods while the adjoint, for the purpose of comparison, is generated by both continuousand discrete methods. Several numerical examples are conducted to demonstrate the effectiveness and accuracyof the proposed methods.

Keywords: Acoustic wave equation, Parameter identification, Finite difference scheme, Inverse problem,Auto differentiation

Inverse Problem for a Structural Acoustic InteractionShitao Liu

Department of Mathematics, University of Virginia, USA.Email: [email protected]

Abstract

In this work, we consider an inverse problem of determining a source term for a structural acoustic partialdifferentia equation (PDE) model, comprised of a two or three-dimensional interior acoustic wave equationcoupled to a Kirchoff plate equation, with the coupling being accomplished across a boundary interface. For thisPDE system, we obtain the uniqueness and stability estimate for the source term from a single measurement ofboundary values of the ”structure”. The proof of uniqueness is based on Carleman estimate. Then, by means ofan observability inequality and a compactness/uniqueness argument, we can get the stability result. Finally, anoperator theoretic approach gives us the regularity needed for the initial conditions in order to get the desiredstability estimate.

Model Function Approach in the Modified L-curve Methodfor the Choice of Regularization Parameter

Shuai LuJohann Radon Institute for Computational and Applied Mathematics,

Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, AustriaEmail: [email protected]

Abstract

In this talk we propose a model function approach in the modified L-curve method for the choice of aregularization parameter. The idea is to replace the residual norm and the regularized solution norm with anapproximated model function. With such an approach, the computational cost of minimum procedure can bedramatically saved. We present numerical tests to support the reliability of the approach. Finally, the modelfunction approach in modified L-curve method is applied to pool boiling data to reconstruct unknown heat fluxesat the boiling surface.

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Optimal Control for the Elliptic System with General ConstraintXiliang Lu

Julius Raab Strasse 10-2538, Linz, A4040, AustriaEmail: [email protected]

Abstract

The optimal control problems governed by partial differential equations with state and/or control constraintsreceived a significant amount of attention recently. The most existing works focus on the unilaterally or bilaterallyconstrained problems. In this talk, we will discuss the optimal control for the elliptic system with the polygonaltype state constraint or point-wise Euclidean norm control constraint. An efficient semi-smooth Newton algorithmis proposed and numerical feasibility of this method is shown.

Some Regularization Methods for the Numerical Analytic ContinuationChu-Li Fu, Xiao-Li Feng, Zhi-Liang Deng, Hao Cheng, Yuan-Xiang Zhang, Yun-Jie MaSchool of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China


Abstract

The problem of analytic continuation of ananalytic function, in general, is an ill-posed problem and it is fre-quently encountered in many practical applications. The main earlier works for this topic focus on the conditionalstability and some rather complicated computational techniques. However, it seems there are few applicationsof the modern theory of regularization methods which have been developed intensively in the last few decades.This paper is devoted to some new regularization methods for solving the numerical analytic continuation ofananalytic function f(z) = f(x+ iy) on a strip domain Ω+ = z = x+ iy|x ∈ R, 0 < y < y0, where the data isgiven approximately only on the line y = 0. Some numerical examples are also provided.

An Adaptive Method for Recovering Sparse Solutionsto Inverse Problems with Applications in System Biology

Jonas OfftermattInstitute of Stochastics and Applications

Department of Optimization, University of StuttgartEmail: [email protected]

Abstract

The approximation of sparse solutions has become an important task in inverse problems settings over thelast few years. This is due to the fact, that many types of functions and signals arising in nature can be describedby a small number of significant degrees of freedom.

In this talk we present an algorithm for cheaply determining sparse solutions of inverse problems. The methodis based on an adaptive discretization scheme, introduced by Ben Ameur, Chavent, and Jaffre for estimating adistributed hydraulic transmissivity. Adaptivity is introduced through refinement and coarsening indicatorswhich allow to efficiently add or remove degrees of freedom. In this talk we discuss a generalization of theseideas, which leads to an algorithm providing sparse solutions.

We applied this algorithm to a recent sparse problem in systems biology, namely the reconstruction of genenetworks out of microarray data sets. This is in general a highly underdetermined and ill-posed problem. Ina first simplified approach, we model it as a linear inverse problem. Taking into account realistic measurement

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3 CONTRIBUTED TALKS

scenarios we arrive at a nonlinear problem. We show numerical results and compare them with other wellknownsparse approximation algorithms.

Quantum Mechanical Inverse Scattering Problemat Fixed Energy: a Constructive Method

Tamas Palmai, Barnabas ApagyiDepartment of Theoretical Physics

Budapest University of Technology and EconomicsBudafoki ut 8., H-1111 Budapest, Hungary


Abstract

The inverse scattering problem of the three-dimensional Schrodinger equation is considered at fixed scatteringenergy with spherically symmetric potentials. Theorems of Loeffel [1], Ramm [2] and Horvath [3] predict theexistence of a unique potential of the class L1,1 from the knowledge of an infinite set of phase shifts (that aredeductable from scattering experiments). Therefore a constructive scheme for recovering the scattering potentialfrom a finite set of phase shifts at a fixed energy is of interest. Such a scheme is suggested by Cox and Thompson[4] and their method is revisited here. A condition is given [5] for the construction of potentials belonging tothe class L1,1 which are the physically meaningful ones. An uniqueness theorem is obtained [5] in the specialcase of one given phase shift by applying the previous condition. An unknown property of the Bessel functionsis discovered [5] in the course of the proof. It is shown that if only one phase shift is specified for the inversionprocedure the unique potential obtained by the Cox-Thompson scheme yields the one specified phase shift whilethe others are small in a certain sense. As the one-phase-shift case occurs frequently in the applications (e.g. atlow energies the s-wave dominates the scattering event) some physical examples are given.

References

[1] J. J. Loeffel, Ann. Inst. Henri Poincare, 8 (1968), pp. 339-447.[2] A. G. Ramm, Commun. Math. Phys., 207 (1999), pp. 231-247.[3] M. Horvath, Trans. Amer. Math. Soc., 358 (2006), pp. 5161-5177.[4] J. R. Cox and K. W. Thompson, J. Math. Phys., 11 (1970), pp. 805-815.[5] T. Palmai and B. Apagyi, J. Math. Phys., 51 (2010), 022114.

Fast Acoustic Imaging for a 3D Penetrable ObjectImmersed in a Shallow Water Waveguide

Wen-Feng Pan1,2,Yun-Xiang You1,Guo-Ping Miao1,Zhuo-Qiu Li21School of Naval Architecture, Ocean and Civil Engineering,

Shanghai Jiaotong University, Shanghai, 2000302School of Science, Wuhan University of Technology, Wuhan, 430070 P.R.China


Abstract

The paper is concerned with the inverse problem for reconstructing a 3D penetrable object in a shallow waterwaveguide from the far-field data of the scattered fields with many acoustic point source incidences. An indicatorsampling method is analyzed and presented for fast imaging the size, shape and location of such a penetrableobject. The method has the advantages that a priori knowledge is avoided for the geometrical and materialproperties of the penetrable obstacle and the much complicated iterative techniques are avoided during theinversion. Numerical examples are given of successful shape reconstructions for several 3D penetrable obstacles

31

3 CONTRIBUTED TALKS

having a variety of shapes. In particular, numerical results show that the proposed method is able to produce agood reconstruction of the size, shape and location of the penetrable target even for the case where the incidentand observation points are restricted to some limited apertures.

On Optimal Reconstruction of Constitutive RelationsV. Bukshtynov1, O. Volkov2, B. Protas2

1School of Computational Engineering and Science, McMaster UniversityHamilton, Ontario L8S 4K1, Canada

2Department of Mathematics and Statistics, McMaster UniversityHamilton, Ontario L8S 4K1, Canada

Emails: [email protected], [email protected], [email protected]

Abstract

In this investigation we develop a computational framework for optimal reconstruction of isotropic constitutiverelationships between thermodynamic variables based on measurements obtained in a spatially-extended system.In other words, assuming the constitutive relation in the following general form[

thermodynamicflux

]= k(state variables)

[thermodynamic

“force”

], (1)

our approach allows us to reconstruct the dependence of the transport coefficient k on the state variables con-sistent with the assumed governing equation(s). Constitutive relations in the form (1) arise in many areas ofnonequilibriumthermodynamics and continuum mechanics. To fix attention, but without loss of generality, inthe present investigation we focus on a heat conduction problem in which the heat flux q represents the thermo-dynamic flux, whereas the temperature gradient ∇T is the thermodynamic “force”, so that relation (1) takes thespecific form

q(x) = k(T )∇T, x ∈ Ω, (2)

where Ω ∈ Rn, n = 1, 2, 3 is the spatial domain on which the problem is formulated. We note that problemsin which the transport coefficient k is a function of the space variable x, rather than the state variable T ,i.e., k = k(x), have received a lot of attention in the literature, and are now relatively well understood. Theoriginality of our contribution consists in that, in contrast to such “parameter estimation” problems, we addressestimation of state-dependent, and therefore nonlinear, constitutive relations. We demonstrate that, as a matterof fact, the mathematical structure of this new problem is quite different from the structure of the parameterestimation problem. Combining constitutive relation (2) with the conservation of energy, we obtain a partialdifferential equation (PDE) describing the distribution of the temperature T in the domain Ω corresponding tothe distribution of heat sources g : Ω → R and suitable boundary conditions (for example, of the Dirichlet type)

−∇ · [k(T )∇T ] = g in Ω, (3a)T = Tb on ∂Ω, (3b)

where Tb denotes the boundary temperature. We note that, for all values of T , we should have k(T ) > 0 whichfollows from the second principle of thermodynamics, but is also required for the mathematical well-posedness ofelliptic boundary value problem (3). The specific inverse problem we address in this investigation is formulatedas follows. Given a set of “measurements” TiM

i=1 of the state variable (temperature) T at a number of pointsxiM

i=1 in the domain Ω, we seek to reconstruct the constitutive relation k(T ) such that solutions of problem(3) obtained with this reconstructed function will fit best the available measurements. An approach commonlyused to solve such inverse problems consists in reformulating them as minimization problems. This is done bydefining the cost functional : R → R as

(k) ,1

2

M∑i=1

[Ti − T (xi; k)]2, (4)

32

3 CONTRIBUTED TALKS

where the dependence of the temperature field T (·; k) on the form of the constitutive relation k = k(T ) is given bygoverning equation (3). The key ingredient of the minimization algorithm is computation of the cost functionalgradient ∇k(k). We emphasize that, since k = k(T ) is a continuous variable, the gradient ∇k(k) represents infact an infinite dimensional sensitivity of (k) to perturbations of k(T ). In our presentation we will show that thisgradient can be determined based on suitably defined adjoint variables and this derivation can be significantlysimplified using the Kirchhoff transformation. These adjoint variables (Lagrange multipliers) are obtained fromthe solution of the corresponding adjoint system which is at the heart of the proposed reconstruction algorithm.Since in general inverse problems often tend to be ill-posed, care must be taken to perform suitable regularization.In our presentation we will review foundations of this approach together with its computational implementation.We will also present a number of computational examples and will discuss extensions of this methods to morecomplicated problems.

Scenario of Seismic Monitoring of Productive ReservoirsGalina Reshetova1, Vladimir Tcheverda2

1Institute of Computational Mathematics and Mathematical Geophysics SB RAS630090 Novosibirsk, Russia

2Institute of Petroleum geology and Geophysics SB RAS630090 Novosibirsk, Russia

Emails: [email protected], [email protected]

Introduction. For over decade 4D or time-lapsing seismic has been effectively applied in reservoir moni-toring during the production life of a series of oil deposits (Clifford et al., 2003). The usual way to do this isimplementation of 3D seismic survey over a producing reservoir. The main objective of this monitoring is tolocate reservoir floodfront and to control flood movement pathways.

Mostly reported successful 4D seismic applications, however, are in offshore field areas where near surfacerelated noise is low and repeatability of seismic signal is rather high. It is necessary to stress that both of theseitems, especially repeatability, are of crucial importance for seismic monitoring. In order to provide the samequality of collected data for land 4D seismic monitoring the reasonable way seems to be in use of acquisitionsystem fixed within a network of boreholes - it provides data with extremely low level of noise and very high rangeof repeatability. Its main disadvantage is possibility to use only few sources/geophones placed rather sparsely.Therefore one should study sensitivity of any particular borehole based acquisition system very carefully beforeto implement it. In order to perform this study, known as survey design, we use Singular Value Decomposition.

Statement of the problem. The essence of seismic monitoring is to follow perturbation of elastic parametersvia inversion of successive data set collected over the same reservoir. Initial distribution ~m0 of these propertiesusually is supposed to be known while its perturbation δ ~m as a rule can be treated as small enough in order tojustify linearization procedure. Therefore seismic monitoring results in resolution of linear operator equation:

DB(~m0) < δ~m >= ~uobscurr − ~uobs

prev (1)

where B(~m0) is nonlinear forward map which transforms model ~m to data ~uobs, DB(~m0) its Frechet derivative,δ ~m perturbation of the model happened between two successive observations - previous and current.

Derivative DB is a compact operator (Khaidukov et al., 1997) and, so, has no bounded inverse. In orderto resolve equation (1) numerically one needs to apply some regularization procedure. In this specific situationr-solution is used (Kostin and Tcheveda, 1995). It is based on truncated SVD for operator DB and consists inresolution of finite-difference version of (1)

Qr(DB(~m0))Pr < δ~m >= Qr(~uobscurr − ~uobs

prev) (2)

with Qr and Pr being orthogonal projectors onto subspaces spanned r left and right singular vectors respectively,corresponding to the largest r singular values.

Singular Value Decomposition and r-pseudoinverse of DB. As was mentioned above for seismicmonitoring borehole based acquisition system is chosen for implementation. It is made from array of sources

33

3 CONTRIBUTED TALKS

placed within producing oil wells (borehole pump acts like a seismic source), while receivers are within non-producing wells.

Singular values computed for this acquisition system is computed and used for analysis of stabilty andresolution ability of the method.

Numerical results of synthetic data inversion are presented and discussed.Acknowledgement. This research was partially supported by RFBR grants 08-05-00265.

Reference:

[1] P. J. Clifford, R. Trythall, R. S. Parr, T. P. Moulds, T. Cook, P.M. Allan, P. Sutcliffe, Integration of4D seismic data into the management of oil reservoirs with horizontal well between fluid contacts, Society ofPetroleum Engineers, SPE No. 83956. Offshore Europe 2003, Aberdeen.[2] V. Khaidukov, V. Kostinand, V. Tcheverda, The r-solution and its application in linearized waveform inversionfor a layered background. In IMA Volume ”Inverse Problems of Wave Propagation”, eds. G. Chavent, W. Symes,1997, Springer, New York, pp. 277-294.[3] V. Kostin, V. Tcheverda, r-pseudoinverse for compact operators in Hilbert spaces: existence and stability, J.Inverse and Ill-posed problems., 1995, 3(2), pp. 131-148.

Travel-time Inversion for 3D Media without of Ray Tracing:Simultaneous Determination of Velocity and Hypocenters

Vladimir A. Tcheverda1, Sergey V. Goldin , Artem V. Kabannik2

1Institute of Petroleum Geology and Geophysics SB RAS, Russia,2Schlumberger Novosibirsk Technological Center, Russia,

Emails: [email protected], [email protected]

Abstract

The paper deals with the problem of simultaneous recovery of a velocity model and hypocenters coordinateby first arrivals recorded by seismic network for local earthquakes. It is the classical problem and a lot of effortsare paid to its resolution, especially for last ten years. But, nevertheless, it is still far away from its completion,especially for 3D statements. In comparison with classical tomographic statement necessity to recover sourcepositions introduces additional troubles and increases level of uncertainty. It should be noted that a posterioriestimation of uncertainty is rather memory and time consuming procedure as it claims repeating data inversionand it is the vulnerability of currently used approaches. We propose to perform inversion iteratively by LSQR- based procedures. Now it is common knowledge that these procedures are regularizing ones with numberof iterations being regularization parameter. This approach opens a possibility to iterative computation of aposteriori estimation of desired parameters together with inversion procedure itself and, so, does not claim anyadditional efforts to do this.First arrivals are treated as an image of some non-linear operator t = B(~h, c) with a composite domain treated as

superposition of two different subsets - finite dimensional set of hypocenters Cartesian coordinate ~h and functionalspace of wave propagation velocity c. Solution of this equation is searched by means of Newton iterative techniquesconsisting in resolution of linear operator equations γ ≡ t−B(~hk, ck) = H < ~hk+1 − ~hk > +A < ck+1 − ck >.These equations are resolved with the help of Pavlis - Booker approach by their splitting on two ”independent”linear equations (see paper Pavlis G.L., Booker J.R. ”The mixed discrete-continuous inverse problem: Applicationto the simultaneous determination of earthquake hypocenters and velocity structure” in J. Geophys. Res.,1980.85, 4801–4810.): [

UT1

UT0

]γ =

[UT

1

0

]< ~hk+1 − ~hk > +

[UT

1

UT0

]A < ck+1 − ck >

where U is matrix made of left singular vectors of H while U1 represents first r of them with r being reasonablychosen rank of H. In order to resolve this equation we start with ck+1 recovery from equation UT

0 γ = UT0 A <

ck+1 − ck >. Its solution is searched by means of LSQR and claims repeated computations of A and AT . We

34

3 CONTRIBUTED TALKS

decline ray tracing procedures to do this but apply finite-difference techniques to numerical resolution of eikonalequation instead. Thus we avoid troublesome procedure of numerical solution of boundary value problem (two-point) ray tracing. Moreover on this way happens to be possible to avoid storage of matrix representation ofoperator A , but just compute its action and action of AT by numerical resolution of two Cauchy problems forthe same linear system of partial differential equations (linearized eikonal equation). On this way computer costof the problem is essentially reduced but still claims parallel implementation, especially for 3D heterogeneousmedia. This approach is tested on synthetic data and applied to inversion of aftershocks data of Chuya 2003year earthquake (Altai region, Russia). Real data are recorded in Altai region (south of Western Siberia) bytemporary and stationary seismic networks of Geophysical Survey of SB RAS.

Keywords: eikonal equation, travel-time inversion, Newton method, iterative regularization, Singular ValueDecomposition.

Numerical Study to the Inverse Source Problemwith Convection -Diffusion Equation

Na Tian1,2,Wenbo Xu1,Choi-Hong Lai21School of Information Technology, Jiangnan University, Wuxi, China

2School of Computing and Mathematical Science, London, UKEmail: [email protected]

Abstract

In this work, a stochastic method named as quantum-behaved particle swarm optimization (QPSO) is usedto solve the inverse source problem associated with transient convection-diffusion equation. This equation isintended to model the process of water pollution. The boundary and initial condition of the concentration areassumed to be known, and then identification of the source term becomes a function specification problem. Thefinite difference method is used as a numerical method for the convection-diffusion equation. The least squaremethod is used to model the inverse problem, which transforms to an optimization problem. Considering thesensitivity to the measurement noise, the Tikhonov regularization method is used to stable the inverse solution.The numerical experiments demonstrate the efficiency and validity of the quantum-behaved particle swarmoptimization. Finally, comparison with a deterministic method known as conjugate gradient method (CGM) isalso presented in this paper.

Tikhonov Regularization with Sparsity Constraints:Convergence Rates and Exact Recovery

Dennis TredeUniversity of Bremen, Germany


Abstract

In this talk we consider linear inverse problems with a bounded linear operator A : H1 → H2 between twoseparable Hilbert spaces H1 and H2,

Af = g. (1)

We are given a noisy observation gε ∈ H2 with ‖g − gε‖H2 ≤ ε and try to reconstruct the solution f of (1) fromthe knowledge of gε.

Moreover, we assume that the operator equation (1) has a solution f† that can be expressed sparsely in anorthonormal basis Ψ := ψii∈N of H1. In [1], the authors introduce the Tikhonov regularization with sparsity

35

3 CONTRIBUTED TALKS

constraints to obtain a sparse approximate solution fα,ε. They define fα,ε as the minimizer of the functional

1

2‖Af − gε‖2H2 + α

∑i∈N

| < f,ψi > |, (2)

with regularization parameter α > 0. In contrast to the classical Tikhonov functional with quadratic penaltythis functional promotes sparsity since small coefficients are penalized stronger.

Stability and convergence rates for the functional (2) have been deduced in [4,5]. For a suitable a prioriparameter choice rule α = α(ε) it has been shown that the Tikhonov functional (2) yields a regularizationmethod. If an additional source condition is fulfilled, then the convergence can be obtained with a linear rate.

This talk deals with stability results for the regularization method consisting in the minimization of (2) andit goes beyond the question of convergence rates. We will deduce an a prior parameter choice rule which ensuresthat the unknown support of the sparse solution f† is recovered exactly, i.e.

support((< fα,ε, ψi >)i∈N) = support((< f†, ψi >)i∈N).

The results which will be presented are a generalization of [3, 8].The practicability of the deduced parameter choice rule will be demonstrated with an examples from digital

holography of particles. Here, the data g are given as sums of characteristic functions convolved with a Fresnelkernel. The measurement of particle size and location amounts to an inverse convolution problem [2, 7].

References:

[1] I. Daubechies, M. Defrise, C. De Mol, An iterative thresholding algorithm for linear inverse problems with asparsity constraint, Communications in Pure and Applied Mathematics, 57(11), pp. 1413-1457, 2004.[2] L. Denis, D. A. Lorenz, E. Thiebaut, C. Fournier, D. Trede, Inline hologram reconstruction with sparsityconstraint, Optics Letters, 34(22), pp. 3475-3477, 2009.[3] J. -J. Fuchs, Recovery of exact sparse representations in the presence of bounded noise, IEEE Transactionson Information Theory, 51(10), pp. 3601-3608, 2005.[4] M. Grasmair, M. Haltmeier, O. Scherzer, Sparse regularization with lq penalty term, Inverse Problems, 24(5):055020 (13pp), 2008.[5] D. A. Lorenz, Convergence rates and source conditions for Tikhonov regularization with sparsity constraints,Journal of Inverse and Ill-Posed Problems, 16(5), pp. 463-478, 2008.[6] D. A. Lorenz, S. Schiffler, D. Trede, Beyond convergence rates: Exact inversion of ill-posed problems withsparsity constraints, Preprint, 2009.[7] F. Soulez, L. Denis, C. Fournier, E.Thiebaut, C. Goepfert, Inverse problem approach for particle digitalholography: accurate location based on local optimisation, Journal of the Optical Society of America A, 24(4):1164-1171, 2007.[8] J. A. Tropp, Just relax: Convex programming methods for identifying sparse signals in noise, IEEE Transac-tions on Information Theory, 52(3), pp. 1030-1051, 2006.

On Inverse Scattering for Nonsymmetric OperatorsIgor Trooshin

Institute of Problems of Precise Mechanics and Control, Russian Academy of SciencesRabochaya 24, Saratov, 410028 Russia


Abstract

We consider a nonsymmetric operator AP in L2(0,∞)2. defined by differential expression

(APu)(x) = Bu′(x) + P (x)u(x), 0 < x <∞

36

3 CONTRIBUTED TALKS

where

B =

(0 11 0

), P (x) =

(p11(x) p12(x)p21(x) p22(x)

),

with the domain

D = u(x) =

(u1(x)u2(x)

)∈ H1(R+)2; u1(0) = hu2(0).

An inverse problem of reconstruction of complex-valued coefficients pij(x) from the scattering data of operatorAP is investigated.

Inverse Problems for Some System ofViscoelasticity via Carleman Estimate

Masaaki UesakaGraduate School of Mathematical Sciences, The University of Tokyo

3-8-1 Komaba Meguro-ku Tokyo 153-8914, JapanEmail: [email protected]

Abstract

The viscoelastic properties of a human body are very important information for a diagonosis of diseases. Inrecent years, a non-invasive method to measure these properties, which is called elastography, is developed. Inthis talk, we will consider a Kelvin-Voigt model, one of the models of viscoelasticity and formulate the inverseproblem for this model to determine the viscoelasticity coefficients from the data on subboundary. We will givethe uniqueness and conditional stability of this inverse problem under several times measurements. The Carlemanestimate which is necessary to prove the stability estimate will be also given.

Error Estimates of Finite Element Methods for ParameterIdentifications in Elliptic and Parabolic Systems

Lijuan WangSchool of Mathematics and Statistics, Wuhan University


Abstract

This work is concerned with the finite element solutions for parameter identifications in second order ellipticand parabolic systems. The L2- and energy-norm error estimates of the finite element solutions are establishedin terms of the mesh size, time step size, regularization parameter and noise level. (This is a joint work with JunZou.)

A FE-based Algorithm for the Inverse Natural Convection ProblemJeff Chak-Fu Wong

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong KongEmail: [email protected]

37

3 CONTRIBUTED TALKS

Abstract

In this talk, several numerical algorithms for solving inverse natural convection problems are revisited andstudied. Our aim is to identify the unknown strength of a time-varying heat source via a set of coupled nonlinearpartial differential equations obtained by the so-called finite element consistent splitting scheme. Viewed as anoptimization problem, the solutions are obtained by means of the conjugate gradient method. A second orderconsistent splitting scheme in time involving the direct problem, the adjoint problem, the sensitivity problem anda system of sensitivity functions is used in order to enhance the numerical accuracy of obtaining the unknownheat source function. A spatial discretisation of all field equations is implemented using mixed and equal-orderfinite element methods. Numerical experiments validate the proposed optimization algorithms that are good anagreement with the existing results.

A Posteriori Error Estimates of Finite Element Methodsfor Heat Flux Reconstructions

Jianli XieDepartment of Mathematics, Shanghai Jiao Tong University


Abstract

We derive a posteriori error estimates of the finite element methods for heat flux reconstructions using theresidual approach. For the stationary inverse problem, we obtain both upper bound and lower bound. It isrevealed for the first time that the upper bound depends explicitly on the regularization parameter. Numericalexperiments are presented to show the applicability and effectiveness of the error estimator.

For the time dependent inverse problem, we obtain a posteriori error estimates in a simple form which canhelp guiding the mesh adaptivity in both space and time.

This is a joint work with Jingzhi Li and Jun Zou.

An Adaptive Greedy Technique for InverseBoundary Determination Problem

F. L. Yang, L. Ling, T. WeiLanzhou University


Abstract

The purpose of this paper is to propose a new numerical method for the inverse boundary determinationproblem of the Laplace equation. The method of fundamental solutions is used for determining an unknownportion of the boundary from the Cauchy data specified on a part of the boundary. Since the resultant ma-trix equation is badly ill-conditioned, the adaptive greedy technique is employed to choose the source points.Meanwhile, the Tikhonov regularization method is used to solve ill-conditional matrix equation, while the regu-larization parameter in the Tikhonov regularization method is provided by the L-curve criterion. The numericalresults show that our proposed method is effective and stable for high noisy data.

38

3 CONTRIBUTED TALKS

Reconstrcting Electromagnetic Obstacles by the Enclosure MethodTing Zhou

Department of Math, University of Washington, USAEmail: [email protected]

Abstract

We study an inverse boundary value problem for Maxwell’s equation. Let Ω ⊂ R3 be a bounded domainwith smooth boundary, filled with isotropic electromagnetic medium, characterized by three parameters: thepermittivity ε(x), conductivity σ(x) and permeability µ(x). A perfect magnetic conducting obstacle is a subsetD of Ω, with smooth boundary, such that the electric-magnetic field (E,H) satisfies the following BVP forMaxwell’s equation

∇∧E = iωµH, ∇∧H = −iω(ε+ i σ

ω

)E in Ω \ øD,

ν ∧E|∂Ω = f,ν ∧E|∂Ω = 0

(0.1)

where ν is the unit outer normal vector to the boundary ∂Ω ∪ ∂D. Define the impedance map by taking thetangential component of the electric field ν ∧ E|∂Ω to the tangential component of the magnetic field ν ∧H|∂Ω.Then our purpose is to retrieve information of the shape of D from the impedance map.

The enclosure method was first introduced by Ikehata [1, 2] to identify obstacles, cavities and inclusionsembedded in conductive or acoustic medium. Geometrically, using the property of so called complex geometricoptics (CGO) solutions that decay on one side and grow on the other side of a hyperplane, one can encloseobstacles by those hyperplanes.

In [3], the Maxewell’s operator was reduced into a matrix Schrodinger operator and vector CGO solutionswere constructed to recover electromagnetic parameters. To address the inverse problem of determining anelectromagnetic obstacle, we observe that solutions of a non-dissipative Maxewell’s equation (σ = 0) sharesimilar asymptotical behavior to those of Helmholtz equations. Therefore, with CGO solutions at hand, theenclosure method is applicable.

References

[1] M. Ikehata, The enclosure method and its applications, Int. Soc. Anal. Comput., 9 (2001), pp. 87-103.[2] M. Ikehata, How to draw a picture of an unkown inclusion from boundary measurements. Two mathematicalinversion algorithms, J. Inv. Ill-Posed Prob., 7 (1999), pp. 255-271.[3] P. Ola, E. Somersalo, Electromagnetic inverse problems and generalized sommerfeld potentials, SIAM. J. Appl.Math., 56 (1996), no. 4, pp. 1129-1145.

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4 LIST OF THE PARTICIPANTS

4 List of the participants

1 Serdyukov Aleksander Novosibirsk State University, Novosibirsk Pirogova str.2 630090, RussiaEmail: [email protected]

2 Mark Asch 75 rue de la Plaine, 75020 Paris, FranceEmail: [email protected]

3 H. T. Banks Center for Research in Scientific Computation, Department of Mathe-matics, North Carolina State University, Raleigh, NC 27695-8205Email: [email protected]

4 Gang Bao Department of Mathematics, Michigan State University, East Lansing,MI 48824-1027Email: [email protected]

5 Hermann Brunner Department of Mathematics, Hong Kong Baptist University, KowloonTong, Hong KongEmail: [email protected]

6 Zhang Cao School of Instrument Science and Opto-Electronics Engineering, Bei-jing University of Aeronautics and Astronautics, Beijing, 100083, P. R.ChinaEmail: zh [email protected]

7 Tony Chan The Hong Kong University of Science and Technology (HKUST)Email: [email protected]

8 Jin Cheng School of Mathematical Sciences, Fudan University, Shanghai, 200433,P. R. ChinaEmail: [email protected]

9 A. Yu. Chebotarev Institute for Applied Mathematics FEB Russian Academy of Science,7 Radio Street, 690041 Vladivostok, RussiaEmail: [email protected]

10 Biquan Chen University of Science and Technology of China, No. 96 Jinzhai Road,Hefei, Anhui, P. R. ChinaEmail: [email protected]

11 Hua Chen School of Mathematics and Statistics, Wuhan University, 430072,Wuhan, P. R. ChinaEmail: [email protected]

12 Yonggang Chen School of Mathematics and statistics , Lanzhou University, 730000,Lanzhou, Gansu, P. R. ChinaEmail: [email protected]

13 Hao Cheng School of Mathematics and Statistics, Lanzhou University, 730000,Tianshui south road 222, Lanzhou, P. R. ChinaEmail: [email protected]

14 Wei Cheng College of Science, Henan University of Technology, 450001, Zhengzhou,P. R. ChinaEmail: [email protected]

15 Ting Cheng Department of Mathematics, Huazhong Normal University, 430079,Wuhan, Hubei, P. R. ChinaEmail: [email protected]

16 Tsz Shun Eric Chung Department of Mathematics, The Chinese University of Hong Kong,Hong KongEmail: [email protected]

40


17 Christian Daveau CNRS (UMR 8088) and Department of Mathematics, University ofCergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-PontoiseCedex, FranceEmail: [email protected], [email protected]

18 ZhiLiang Deng School of Mathematics and Statistics, Lanzhou University, 730000,Tianshui south road 222, Lanzhou, P. R. ChinaEmail: [email protected]

19 Francois-XavierLe Dimet

Lab Jean-Kuntzman, Universite Joseph Fourier, BP 53, 38041 Grenoblecedex 9, FranceEmail: [email protected]

20 FangFang Dou School of Applied Mathematics, University of Electronic Science andTechnology of China, 610054, Chengdu, Sichuan Province, P. R. ChinaEmail: [email protected]

21 Yixin Dou Department of Mathematics, Harbin Institute of Technology, Harbin,Heilongjiang, 150001, P. R. ChinaEmail: [email protected]

22 Heinz W. Engl University of Vienna, Johann Radon Institute for Computational andApplied Mathematics (RICAM), Austrian Academy of Sciences, A-4040Linz, AustriaEmail: [email protected]

23 Elisabeth Engl Altenberger Strasse 69, 4040 Linz, AustriaEmail:

24 Hui Feng School of Mathematics and Statistics, Wuhan University, Wuhan,430072, P. R. ChinaEmail: [email protected]

25 Lixin Feng Mathematics Department, Heilongjiang University, Harbin, Hei-longjiang, 150080, P. R. ChinaEmail: [email protected]

26 Chuli Fu School of Mathematics and Statistics, Lanzhou University, Tianshuisouth road 222, Lanzhou, 730000, P. R. ChinaEmail: [email protected]

27 Wenqiang Feng Department of Mathematics, University of Science and Technology ofChina, No. 96 Jinzhai Road, Hefei, Anhui, P. R. ChinaEmail: [email protected]

28 Adel Hamdi 14 rue pavee, appartement 40, 76100 Rouen, FranceEmail: [email protected]

29 Guanghui Hu Weierstrass Institute for Applied Analysis and Stochastics (WIAS),Mohrenstr. 39, 10117 Berlin, GermanyEmail: [email protected]

30 Baoqing Hu School of Mathematics and Statistics, Wuhan University, Wuhan,430072, P. R. ChinaEmail: [email protected]

31 Hui Huang Department of Mathematics and Earth and Ocean Sciences, Univer-sity of British Columbia, 221-2875 Osoyoos Crescent, Vancouver, BC,V6T2G3, CanadaEmail: [email protected]

32 Victor Isakov Department of Mathematics and Statistics, Wichita State University,Wichita, Kansas 67260-0033, USAEmail: [email protected]

41


33 Kazufumi Ito Department of Mathematics, North Carolina State University, Raleigh,North Carolina, USAEmail: [email protected]

34 Daijun Jiang School of Mathematics and Statistics, Wuhan University, Wuhan,430072, P. R. [email protected]

35 Bangti Jin Center for Industrial Mathematics, University of Bremen, D-28334,Bremen, GermanyEmail: [email protected]

36 Vincent Jugnon Centre de Mathematiques Appliquees, CNRS UMR 7641, Ecole Poly-technique, 91128 Palaiseau, FranceEmail: [email protected]

37 Sergey I. Kabanikhin Sobolev Institute of Mathematics, Siberian Branch of Russian Academyof Science, Novosibirsk, RussiaEmail: [email protected]

38 Hyeonbae Kang Department of Mathematics, Inha University, Incheon 402-751, S. Ko-reaEmail: [email protected]

39 Rainer Kress Institut fur Numerische und Angewandte Mathematik, UniversitatGottingen, GermanyEmail: [email protected]

40 Philipp Kuegler Johann Radon Institute for Computational and Applied Mathematics,AustriaEmail: [email protected]

41 Karl Kunisch Institute of Mathematics, Scientific Computing, University of Graz,Heinrichstrasse 36, A-8010 Graz, AustriaEmail: [email protected]

42 Dingfang Li School of Mathematics and Statistics, Wuhan University, Wuhan,430072, P. R. ChinaEmail: [email protected]

43 Tatsien Li School of Mathematical Sciences, Fudan University, Shanghai, 200433,P. R. ChinaEmail: [email protected]

44 Jianliang Li Academy of Mathematics and Systems Science, Chinese Academy ofScience, No. 55, East Road of Zhongguan cun, Haidian District, Bei-jing, 100190, P. R. ChinaEmail: [email protected]

45 Jingzhi Li HG G 55, Seminar for Applied Mathematics, D-Math, Swiss FederalInstitute of Technology Zurich, Ramistrasse 101, CH-8092, Zurich,SwitzerlandEmail: [email protected]

46 Shumin Li Department of Mathematics, University of Science and Technology ofChina, Hefei, Anhui Province, 230026, P. R. ChinaEmail: [email protected]

47 Weiguo Li School of mathematical and computational sciences, China Univer-sity of Petroleum, 271 Beier Road, Dongying City, 257061, ShandongProvince, P. R. ChinaEmail: [email protected]

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48 Yuan Li Mathematics Department, Heilongjiang University, Harbin, Hei-longjiang, 150080, P. R. ChinaEmail: [email protected]

49 Zhiyuan Li Department of Mathematics, University of Science and Technology ofChina, Hefei, Anhui Province, 230026, P. R. ChinaEmail: [email protected]

50 Wenyuan Liao Department of Mathematics and Statistics, University of Calgary, 2500University Drive, NW, Calgary, Alberta, T2N 1N4, CanadaEmail: [email protected]

51 Hao Liu College of Science, Nanjing University of Aeornautics and Astronautics,Nanjing, P. R. ChinaEmail: [email protected]

52 JiChuan Liu School of Mathematics and Statistics, Lanzhou University, Tianshuisouth road 222, Lanzhou, 730000, P. R. ChinaEmail: [email protected]

53 Jijun Liu Department of Mathematics, Southeast University, Nanjing, 210096,P.R.ChinaEmail: [email protected]

54 Shitao Liu P.O.Box 400137, Kerchof Hall, Department of Mathematics, Universityof Virginia, Charlottesville, VA 22904, USAEmail: [email protected]

55 Xiaodong Liu Academy of Mathematics and Systems Science, Chinese Academy ofSciences, No.55, East Road of Zhongguan cun, Haidian District, Beijing100190, P. R. ChinaEmail: [email protected]

56 Shuai Lu Johann Radon Institute for Computational and Applied Mathematics,Altenbergerstrasse 69, A-4040 Linz, AustriaEmail: [email protected]

57 Xiliang Lu Julius Raab Strasse 10-2538, Linz, A4040, AustriaEmail: [email protected]

58 YunJie Ma School of Mathematics and Statistics, Lanzhou University, Tianshuisouth road 222, Lanzhou, 730000, P. R. ChinaEmail: [email protected]

59 Dong Miao Department of Mathematics, Nanjing University, No. 22 HanKouStreet, Nanjing 210093, P. R. ChinaEmail: [email protected]

60 Peter Monk Department of Mathematical Sciences, University of DelawareEmail: [email protected]

61 Wuqing Ning Department of Mathematics, University of Science and Technology ofChina, Hefei, Anhui Province, 230026, P. R. ChinaEmail: [email protected]

62 Jonas Offtermatt Institute of Stochastics and Applications, University of Stuttgart, Pfaf-fenwaldring 57, 70569 Stuttgart, GermanyEmail: [email protected]

63 Tamas Palmai Department of Theoretical Physics, Budapest University of Technologyand Economics, Budafoki ut 8., H-1111 Budapest, HungaryEmail: [email protected]

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64 Wenfeng Pan School of Science, Wuhan University of Technology, 122 Luoshi Road,Wuhan, 430070, P. R. ChinaEmail: [email protected]

65 Roland Potthast German Meteorological Service - Deutscher Wetterdienst, Research andDevelopment, Leader Section FE 12 (Data Assimilation), FrankfurterStrasse 135, 63067 Offenbach, GermanyEmail: [email protected]

66 Bartosz Protas Department of Mathematics and Statistics, McMaster University, 1280Main Street West, Hamilton, Ontario L8S 4K1, CanadaEmail: [email protected]

67 Ailin Qian School of Mathematics and Statistics, Lanzhou University, Lanzhou,730000, P. R. ChinaEmail: [email protected]

68 Zhi Qian Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. ChinaEmail: [email protected]

69 Galina Reshetova Institute of computational mathematics and mathematical geophysic-spr. Lavrentieva, 6630090, NovosibirskRussiaEmail: [email protected]

70 William Rundell Department of Mathematics, Texas A&M University, College Station,Tx 77843-3368Email: [email protected]

71 Guanying Sun Academy of Mathematics and Systems Science, Chinese Academy ofScience. No.55, East Road of Zhongguan cun, Haidian District, Beijing,100190, P. R. ChinaEmail: [email protected]

72 Hongpeng Sun Academy of Mathematics and Systems Science, Chinese Academy ofScience. No.55, East Road of Zhongguan cun, Haidian District, Beijing,100190, P. R. ChinaEmail: [email protected]

73 VladimirA. Tcheverda

Institute of Petroleum Geology and Geophysics. SB RAS, 3, propsp.Koptyug, 630090, Novosibirsk, RussiaEmail: [email protected]

74 Na Tian School of Information Technology, Jiangnan University, Wuxi, 214082,P. R. ChinaEmail: [email protected]

75 Dennis Trede Zentrum fur Technomathematik, Universitat Bremen, Bibliothekstrasse2, D-28195 Bremen, GermanyEmail: [email protected]

76 Igor Trooshin 4-16-3 Kugenuma-kaigan, Fujisawa-shi, Kanagawa, 251-0037 JapanEmail: [email protected]

77 Masaaki Uesaka Graduate School of Mathematical Sciences, The University of Tokyo,3-8-1 Komaba Meguro-ku Tokyo 153-8914, JapanEmail: [email protected]

78 Gunther Uhlmann Department of Mathematics, University of Washington, C-449Padelford Hall Box 354350 Seattle, Washington 98195-4350 USAEmail: [email protected]

79 Gengsheng Wang Department of Mathematics, School of Mathematics and Statistics,Wuhan University, Wuhan, 430072, P. R. ChinaEmail: [email protected]

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80 Jenn-Nan Wang Department of Mathematics, National Taiwan University, Taipei 106,TaiwanEmail: [email protected]

81 Lijuan Wang School of Mathematics and Statistics, Wuhan University, Wuhan,430072, P. R. ChinaEmail: [email protected]

82 Rong Wang School of Mathematics and Statistics, Wuhan University, Wuhan,430072, P. R. ChinaEmail: [email protected]

83 Jin Wen School of Mathematics and Statistics, Lanzhou University, 730000,Lanzhou, Gansu, P. R. ChinaEmail: [email protected]

84 Jeff Chak-Fu Wong Department of Mathematics, The Chinese University of Hong Kong,Shatin, Hong KongEmail: [email protected]

85 Hua Xiang School of Mathematics and Statistics, Wuhan University, Wuhan,430072, P. R. ChinaEmail: [email protected]

86 Jianli Xie Department of Mathematics, Shanghai Jiao Tong University, Shanghai,200240, P. R. ChinaEmail: [email protected]

87 Yuanming Xiao Nanjing University, 22 Hankou Road, Nanjing, Jiangsu 210093, P. R.China.Email: [email protected]

88 Dinghua Xu Department of Mathematics, College of Sciences, Zhejiang Sci-TechUniversity, Hangzhou, Zhejiang Province, 310018, P. R. ChinaEmail: [email protected]

89 Yuesheng Xu Illinois Institute of Tech, Syracuse University and Sun Yat-sen Univer-sityEmail: [email protected]

90 Masahiro Yamamoto Department of Mathematical Sciences, University of Tokyo, 3-8-1Komaba Meguro Tokyo 153, JapanEmail: [email protected]

91 Guozheng Yan Department of Mathematics, Huazhong Normal University, 430079,Wuhan, Hubei, P. R. ChinaEmail: yan [email protected]

92 Fenglian Yang School of Mathematics and Statistics, Lanzhou University, Lanzhou,730000, P. R. ChinaEmail: [email protected]

93 Jiaqing Yang Academy of Mathematics and Systems Science, Chinese Academy ofSciences, No.55, East Road of Zhongguancun, Haidian District, Beijing,100190, P. R. ChinaEmail: [email protected]

94 Fangman Zhai Nanjing Forestry University, No. 159, Longpan Road, Nanjing, 210037,Jiangsu, P. R. ChinaEmail: [email protected]

95 Bo Zhang LSEC and Institute of Applied Mathematics, Academy of Mathematicsand Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. ChinaEmail: [email protected]

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96 Haiwen Zhang Academy of Mathematics and Systems Science, Chinese Academy ofSciences, No.55, East Road of Zhongguancun, Haidian District, Beijing,100190, P. R. ChinaEmail: [email protected]

97 Ran Zhang Department of Mathematics, Jilin University, Changchun, 130012, P.R. ChinaEmail: [email protected]

98 Xu Zhang Academy of Mathematics and Systems Science, Chinese Academy ofSciences, Beijing, P. R. ChinaEmail: [email protected]

99 Yuanxiang Zhang School of Mathematics and Statistics, Lanzhou University, Tianshuisouth road 222, Lanzhou, 730000, P. R. ChinaEmail: [email protected]

100 Chunxiong Zheng Department of Mathematical Sciences, Tsinghua University, Beijing,100084, P. R. ChinaEmail: [email protected]

101 Huanlin Zhou Department of Engineering Mechanics, Hefei University of Technology,No.193, Tunxi Road, Hefei, Anhui, P. R. ChinaEmail: huanlin [email protected]

102 Ting Zhou Dept. of Math, University of Washington, 4725 24th Ave NE APT405,Seattle, WA 98105, USAEmail: [email protected]

103 Zhixiong Zong School of Science, Wuhan University of Technology, 122 Luoshi Road,Wuhan, 430070, P. R. ChinaEmail: [email protected]

104 Jun Zou Department of Mathematics, The Chinese University of Hong Kong,Shatin, N. T., Hong KongEmail: [email protected]

105 Xiufen Zou School of Mathematics and Statistics, Wuhan University, Wuhan,430072, P. R. ChinaEmail: [email protected]

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5 DIRECTION INFORMATION

5 Direction Information

Directions for International Conference on Inverse Problems

All participants are arranged to stay at the conference hotel:FengYi Hotel (´ ËA), Wuhan (ÉÇ), China (¥I)

Hotel address: No. 336, Bayi Road, Wuchang, Wuhan, Hubei province (ÉÇ½Él´336Ò )Telephone: 0086-27-67811888

1. From Wuhan Tianhe International Airport (ÉÇUàISÅ|) to FengYi Hotel

On April 25, 2010, Sunday, from 9:00am-9:00pm:

Find some conference staff with signs for ”International Conference on Inverse Problems”, you willbe guided to either the airport taxis or the airport shuttle buses.

Taxis:You may directly take a taxi from the airport to FengYi Hotel (´ ËA), it takes about 45-60 minutes

(approx. CNY110).

Airport shuttle buses:The airport shuttle bus runs once every 30 minutes, from 9:00am to the last flight of each day. You should

get off the shuttle bus at Fujiapo Station (G[·Õ), Wuchang (it costs CNY30). Then you may take a taxifrom Fujiapo Station (G[·Õ) to FengYi Hotel (´ ËA) (approx. CNY15).

2. From Railway Stations to FengYi Hotel

You may take a taxi from the Wuchang Railway Station, Hankou Railway Station or Wuhan Rail-way Station to Fengyi Hotel (´ ËA). It takes about 15, 45 and 30 minutes respectively (approx. CNY18,40 and 35 resp.)

3. A useful Chinese sentence

Show this sentence to your taxi or bus driver if needed in Wuhan:

x·´ ËA£ÉÇ½Él´336Ò. >: 67811888), u¦.(Please drop me off at FengYi Hotel, and give me a receipt of the taxi fare)

4. Contact persons and phone numbersYou may call the following cell phone numbers when you need assistance in China:

13657277240 or 63625372 (Prof. Lijuan Wang)13397177528 (Prof. Xiufen Zou)13986110652 (Prof. Hui Feng)15927510187 (Mr. Daijun Jiang)

5. The map of FengYi Hotel

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5 DIRECTION INFORMATION

48

international conference on inverse problems

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